Let me first compare some of the verses of Valmiki Ramayana among themselves and other verses of Mahabharata. Then I will give my conclusion. I know it's a bit long but please read till the end.
The below verses are from Valmiki Ramayana – Bala Kanda – Chapter 75. These is a word to word translation given by sri Desiraju Hanumanth Rao ji who had translated the whole Valmiki Ramayana into english. He translates that verse as follows.
ime dve dhanuSii shreSThe divye loka abhipuujite
dR^iDhe balavatii mukhye sukR^ite vishvakarmaNaa 1-75-11
“These are the two strong and sturdy unsurpassed longbows, well-designed by gods and well-crafted by Vishvakarma, the Divine Architect, and these are very important among all bows and well-worshipped by all worlds… one broken in your, and the other in my hand…” [1-75-11]
anisR^iSTam suraiH ekam tryambakaaya yuyutsave
tripura ghnam narashreSTha bhagnam kaakutstha yat tvayaa 1-75-12
“Oh, best one among men, out of the two longbows gods gave one to restive Trymbaka, God Shiva for a combat with demon Tripura, and oh, Kakutstha, that bow alone is the annihilator of Tripura, the demon… and you have broken that alone…” [1-75-12]
adaa tu jR^imbhitam shaivam dhanuH bhiima paraakramam 1-75-17
hum kaareNa mahaadevaH stambhito atha trilocanaH
“By the ‘hum’ sound of Vishnu that ruinously overpowering longbow of Shiva is broken (unstrung), and the triple-eyed God, Mahadeva, is frozen…” [1-75-17b, 18a]
dhanuu rudraH tu sa.mkruddho videheSu mahaayashaaH 1-75-20
devaraatasya raaja R^iSeH dadau haste sa saayakam
“That celebrated Rudra on his part with indignation has handed over that longbow, which is already fitted with unloosened arrow, to the sagely king among Videha kings, namely Devaraata…” [1-75-20b, 21a]
Note carefully .: "Here in this verses it is said that Lord Vishwakarma had made the bow for lord Shiva which he used to destroy tripurasura, was broken or unstrung by the "hum" of Lord vishnu and he gave this bow (with already arrow fitted with unloosened arrows) directly to king Devaraata."
In these verses only, there are many uncertainties let me point one rest I will compare with other verses. "If it was untrung by the hum than how it can have already unloosened arrow fitted in it.?"
No one can question the authenticity and originality of these translation here because it's given by sri Desiraju Hanumanth Rao ji and I have given word to word translation here. Now let's move forward.
Mahabharata has a detailed account of the story of Tripura Samhaara (Destruction of three cities) by lord Shiva. Vyasa himself narrates that story to Arjuna whose excerpt has been copied below. This excerpt has been taken from the Kisari Mohan Ganguli’s translation of the Vyasa Bharata.
Book-7: Drona Vadha Parva:SECTION CCII
“Mahadeva thus accepted their request, moved by the desire of benefiting them, and said, ‘I will overthrow these Asuras. And Hara made the two mountains, viz., Gandhamadana and Vindhya, the two poles of his car. And Sankara made the earth with her oceans and forests his battle car. And the three-eyed deity made that prince of snakes, viz., Sesha, the Aksha, of that car. And that God of gods, the wielder of Pinaka, made the moon and the sun the two wheels of that vehicle. And the triple-eyed Lord made Elapatra and Pushpadanta, the two pins of the yoke. And the valiant Mahadeva made the Malaya mountains the yoke, and the great Takshaka the string for tying the yoke to the poles, and the creatures about him the traces of the steed. And Maheswara made the four Vedas his four steeds. And that lord of the three worlds made the supplementary Vedas the bridle-bits. And Mahadeva made Gayatri and Savitri the reins, the syllable Om the whip, and Brahma the driver. And making the Mandara mountains the bow, Vasuki the bowstring, Vishnu his excellent shaft, Agni the arrow-head, and Vayu the two wings of that shafts, Yama the feathers in its tail, lightning the whetting stone, and Meru the standard, Siva, riding on that excellent car which was composed of all the celestial forces, proceeded for the destruction of the triple city”.
This story has been narrated by Veda Vyasa himself, and exists in Mahabharata which is an EPIC. Therefore we have two noteworthy plus-points to consider this story as authentic, one is Vyasa’s spoken words, and second is presence in Epic Mahabharata.
This story clearly contradicts the points summarized above. Here we have clearly seen that there is no Viswakarma involved in creation of any bow. Mahadeva himself did his preparation. Note that Shiva made the mountain range of “Mandara” his divine bow to destroy the triple cities.
The story of bestowal of Shiva’s bow to the dynasty of Janaka is said variously at various places. This bow obtaining story is itself NOT consistent.
1. In the hermitage of Vishvamitra it is said that the bow is given in Vedic-ritual
“taddhi puurvam narashreSTha dattam sadasi daivataiH 1-31-8”
2. In the chapter 75 of Bala Kanda as seen in above verses, it is said that the bow used in Tripura Samhaara was given in the hand of Devaraata
“devaraatasya raaja R^iSeH dadau haste sa saayakam 1-75-20”
3. And in chapter 66 of Bala Kanda it is said that the bow which Shiva used to destroy Daksha’s sacrifice, that was the same bow which was given in Dasksha’s ritual to Gods.
“dakSa yaj~na vadhe puurvam dhanuH aayamya viiryavaan 1-66-9”
4. Seetha says about this to ascetic Lady Anasuya that Rain-god gave this to Janaka’s dynasty
“mahaayaj~ne tadaa tasya varuNena mahaatmanaa dattam dhanur varam priityaa tuuNii ca akSayya saayakau Ayodhya II-118”(as mentioned in the question itself).
We can combine quotes no. # 3 & # 4 and frame a thought that the bow which was used by Shiva to destroy Daksha’s sacrifice was the same given to Rain god and Rain god Passed that on to may be Devaraata. Then these points # 3 & # 4. contradict point no. # 2.
Definitely Daksha Yajna bow is NOT SAME as Tripura Samhaara Bow since for the destruction of three cities the bow was specially made then and there and was NOT any bow used previously so far.
Another CRITICAL point to be noted is, we also can NOT assume that after destroying Tripura Shiva kept that bow with himself and later in Daksha sacrificed used the same bow and may be sometime later fought with Vishnu and got defeated and gave away that bow to Devaraata through rain god. This is also NOT A VALID assumption because, Daksha Sacrifice happened when Shiva married Sati (Shiva’s first consort), and Shiva destroyed Tripura after the birth of Kartikeya who was born after Shiva married Parvati (second incarnation). Kartikeya slayed Tarakasura and the Tripura owners (demons) were the three sons of Tarakasura.
Therefore, the Daksha Sacrifice incident precedes to Tripura destruction incident in sequence of time. So, that means Shiva definitely used a different bow in Daksha sacrifice which he might have gifted to rain God and he inturn gave that to Devaraata. And later in Tripura Samhaara, Shiva created a new bow using mandara mountain which was called Pinaka. Which lord shiva used during his battle with lord krishna in the banasura episode and also during his battle with arjuna during kirat Parva in Mahabharata.
So the fight between Lord Shiva and Lord Vishnu kind of contradicts rest of the verses and Mahabharata. Still let's visit those verses again with a new angle.
"Because of Rudra’s raudra (angry) form, Lord Vishnu uses the hunkārā sound (hum, which is Om) to calm him. Hearing that sound(nAda), Rudra goes into meditation. Thus his bow becomes jrambhita(frozen strung with arrow as it is) & Rudra becomes stambhita, going into samAdhi."
Lord Shiva is the humkara himself so called Mahakala. Again the sound "hum" is equivalent to "aum" Naada. Even in Buddhism "hum" is used for "aum". (Now just don't say it's another different religion as it's originated from hinduism only like Sikhism and Jainism also Gautama Buddha is said to be the 9th avatar of Lord Vishnu among the dasha avataras).
Heart Mantra(Suragami Dharini).
"The Heavenly One! AUM!
He proceeds on the wind like fire, manifesting His devotion in place after place! The Diamond-Thunderbolt Handed One! HUM! To Him of Universal Attributes, Hail!"
Plus according to Mahabharata.: the below verses from the Drona Parva of the Mahabharata and the Harivamsa reveal the boon given by lord shiva to lord vishnu after 66000 celestial years of penance as thus:
matprasādān manuṣyeṣu devagandharvayoniṣu
aprameyabalātmā tvaṃ nārāyaṇa bhaviṣyasi
( Mahabharata 7 : 172: verse No. 74 )
By the grace of me you will become Nārāyaṇa with unending strength among the deities, humans and Gandharvas
kaś cit tava rujaṃ kartā matprasādāt kathaṃ cana
api cet samaraṃ gatvā bhaviṣyasi mamādhikaḥ
( Mahabharata 7 : 172: verse No. 78 )
“Through my grace, none shall ever be able to cause thee pain by the weapon of thunderbolt or with any object that is wet or dry, or with any mobile or immobile thing. Thou shalt be superior to myself if thou ever goest to battle against me.'”
This was again confirmed by Lord Siva to Lord Krishna before Lord Krishna was about to appear in battle against Vanasura as stated in the Harivamsa :
yathA mainAkamAshritya tapastvamakaroH prabho |
tathA mama varaM kR^iShNa saMsmR^itya sthairyamApnuhi ||2-74-37
O lord ! O kR^iShNa! As you did penance staying on mainAka, you received a boon from me. Be firm in your mind, remembering that boon.
avadhyastvamajeyashcha mattaH shUratarastathA |
bhavitAsItyavochaM yattattathA na tadanyathA || 2-74-38
You can not be killed, you can not be conquered, you will be more valiant than me. All this will happen as told by me. None will be able to change this.
Otherwise, who else is able to dare appear before Lord Siva in the battlefield when He holds His trident?
So it was Lord Shiva only who gave Lord Vishnu the boon of remain undefeated even against him.
This suits it more otherwise it will contradict vedas also which states.:
"imā rudrāya sthiradhanvane ghiraḥ kṣipreṣave devāya svadhāvne |
aṣāḷhāya sahamānāya vedhase tighmāyudhāya bharatā śṛṇotu naḥ |” (Rig Veda 7:46:1)
“To Rudra bring these songs, whose bow is firm and strong, the self-dependent GOD with swiftly-flying shafts, The Wise, the Unconquered Conqueror whom none may overcome, armed with sharp-pointed weapons: may he hear our call. “
“śreṣṭho jātasya rudra śriyāsi tavastamastavasāṃ vajrabāho |
parṣi ṇaḥ pāramaṃhasaḥ svasti viśvā abhītī rapaso yuyodhi |” (Rig Veda 2.33.3)
“In beauty thou art the most beautiful of all that exists, O Rudra, the strongest of the strong, thou wielder of the thunderbolt! Carry us happily to the other shore of our anguish, and ward off all assaults of mischief”.
“arhan bibharṣi sāyakāni dhanvārhan niṣkaṃ yajataṃ viśvarūpam |
arhannidaṃ dayase viśvamabhvaṃ na vā ojīyo rudra tvadasti |” (Rig Veda 2:33:10)
“Worthy, thou carriest thy bow and arrows, worthy, thy manyhued and honoured necklace.Worthy, thou cuttest here each fiend to pieces: a mightier than thou there is not, Rudra“.
“namah sutayahantyaya |” (Yajurveda Sri Rudram – Anuvaka-2)
“Salutations to Lord Rudra who cannot be overcome (defeated) and slain”.
“Namah sahamanaya nivyadhina avyadhininam pataye namo |” (Yajurveda Sri Rudram, Anuvaka-3)
“Salutations to Him who can not only withstand the shock of the onset of His enemies, but overpower them. He who can effortlessly pierce His enemies; the Lord of those who can fight on all sides, salutations to Him”.
Conclusion.: The bow is definitely shiva dhanush given by lord Varuna to king Devaraata as all the weapons are said to be lord Shiva's only who gave them to other deities like vajra, asi, sudarshana, parshu, 3 baans, etc. But it's not his own bow "Pinaka" it's just another one of Shiva dhanusha like the one lord Krishna broke in kansa's ranga sabha in Mathura.