The question itself indicates that the questioner had done some deeper study of Ramayana and in his/her post script, the questioner called for a well-researched answers.
According to my study I had understood that
a) Bala Kanda contains major interpolations
I. Prologue to the story
The first 4 Sargas of Bala Kanda. It is in these Sargas we will come across the stories of
Sage Narada describing the great qualities of Sri Rama
Lord Brahma coming down personally to the Ashram of Sage Valmiki and ordering him to
pen the Srimad Ramayana.
Singing of Ramayana by Lava and Kusa.
(1) Ramayana was written at the Vedic age. Sage Narada was not a part of Vedic Literature. Hence, his coming and explaining about Sri Rama is an insertion made at a later date.
(2) All Vedas were heard by Rishies in Dhyana and they gave that knowledge to the world. In no case we heard that Lord Brahma himself coming down and directing a sage to share the Vedic Knowledge with the world. So the story of Lord Brahma ordering Sage Valmiki is an INTERPOLATION.
(3) The Story of Ramayana ends with the coronation of Sri Rama as the king of Kosala Kingdom. It was only in the PRAKSHIPTA KANDA i.e, Uttara Kanda Lava and Kusa were mentioned. Hence, their presence in Baala Kanda does not fit in.
So the first 4 Sargas of Baala Kanda are interpolation with a view to elevate the status of Vishnu to the level of SUPREME GOD.
II. Conduct of Putrakaamesthi Yagna
King Dasaratha never thought of conducting of Putrakaamesthi Yagna. This story is incorporated at a later date (interpolation) in 15 - 17 sargas (3 in number ) to elevate the status of Vishnu to that of SUPREME God and to establish that Sri Rama was an incarnation of Vishnu.
III) The story of Parasurama
Parasurama was not an incarnation of Vishnu. This story was incorporated in 75 - 77 Sargas (3 Sargas) of Baala Kanda. This story was incorporated with a view to establishing the 6th incarnation of Vishnu, an interpolation.
IV) Sargas/Cantos that are not in tune with Sage Valmiki's style
If we read Srimad Ramayana carefully, we will find that Sage Valmiki deviated sparingly from the Main story. If it all he had to narrate about a sub-story, he used to describe that sub-story in brief. It is his style of narrating the story.
For example: Whether it is (i) describing the greatness of Anasuya, the wife of Sage Atri in Ayodhya Kanda, (ii) narrating the greatness of Sage Agastya in Aranya Kanda, (iii) telling about the enmity between Vali and Sugreeva in Kishkinda Kanda, (iv) the curse imposed on Kumbakarna by Lord Brahma in Yuddha Kanda, etc, we will observe his style of narrating the sub-story in brevity.
Contrary to this style, we will find many Sargas/Cantos in Baala Kanda of Srimad Ramayana.
- Bringing Sage Rishyasringa to Ayodhya.
In order to perform Aswamedha Yagna Sage Rishyasringa was invited and brought to Ayodhya. However, apart from this brief episode, the story of Rishyasringa was narrated in 4 sargas (9-11) in Baala Kanda. As it is not no related to the main story, we have to omit these 4 Sargas.
- The stories of Ganga, Sagara, Ahalya, etc
The stories of Ganga, Uma, the birth of Kaartikeya, the story of Sagara, digging of Earth by the sons of Sagara, the story of Bhagiratha, Ahalya, etc, were described from Sargas 35 - 50 (16 in number) in Baala Kanda. These stories have no relevance to the story of Sri Rama,i.e, Ramayana. Hence, these are interpolations.
- The story of Viswamitra
The story of Sage Viswamitra was narrated from 51 to 65 Sargas (15 in number) in Baala Kanda. This story has no relevance to the story of Sri Rama,i.e, Ramayana. Hence, these are interpolations.
Thus we have interpolations in 45 Sargas out of 77 sargas in Bala Kanda.
b) Ayodhya, Aranya, Kishkinda and Sundara Kandas contains interpolations here and there.
1) Sri Rama gives instructions to Bharata about ruling of kingdom. It is quite peculiar as to why Sri Rama should start teaching about ruling after a re-unite with Bharata. It is not Valmiki's style.
2) Sri Hanuman searches for Sita in Lanka and sees Pushpaka. Here the following Sarga appears to be interpolation, as it contains description Lakshmi as the Goddess being venerated by elephants, which is a Puranic Concept but not Vedic. In Vedic era Lakshmi means riches.
नियुज्यमानाः च गजाः सुहस्ताः |
सकेसराः च उत्पल पत्र हस्ताः |
बभूव देवी च कृता सुहस्ता |
लक्ष्मीः तथा पद्मिनि पद्म हस्ता || ५-७-१४
Images of elephants in a lotus-pool with filaments of lotus on body, with lotus petals held in their trunks, were devoted to the worship of an image of Goddess Lakshmi. And also an image of Goddess Lakshmi with four graceful hands and holding lotus in Her hand was made to exist in Pushpaka.
c) The Sita Parityaga (dis-owning of Sita by Sri Rama) episode, consequent Agni Pravesha (jumping into pyre by Sita) episode, arrival of Brahma, Shiva and other gods episode are interpolations. This was already discussed by me at the following link:
Why did Lord Rama ask mother Sita for Agni Pariksha?
d) Uttara Kanda is an interpolated story in toto. It was already discussed by me at the following link:
Why did Sri Rama banish Lady Sita when he was a witness of her fire-test himself?
Sage Valmiki gave the Ramayana to the World, to present a role model to human beings in the form of Sita and Sri Rama. It is naive to think that he wrote his poem in 24,000 slokas, though ardent devotees of Gayatri may differ with me.
The Ramayana story starts with Sarga 5 of Baala Kanda. We have to add the remaining 32 Sargas to Ayodhya Kanda, making it as a single Kanda.
So we have only 5 Kandas in the Srimad Ramayana.
A) According to the Critical edition of Vālmīki Rāmāyaṇa , the total number of ślokas is only 18,670. Out of which 100 sargas consisting of 2,689 Slokas pertain to Uttara Kanda (as per questioner's answer).
B) As my study proved that Uttara Kanda is an interpolation in toto, we have to delete 2,689 slokas of Uttara Kanda, for arriving at a realistic number of slokas in Ramayana. So it leaves us with 15,981 slokas (18,670 - 2,689).
C) Further, I had also mentioned above that 45 sargas in Bala Kanda are interpolations. On an average there will be 20 slokas in a Sarga. So we have to delete 900 slokas (45 sargas x 20 slokas). This will bring down the actual slokas in Ramayana to 15,081 (15,981 - 900).
D) If we consider minor interpolations in Ayodhya, Aranya, Kishkinda and Sundara Kandas we may arrive at actual slokas in Ramayana to be below 15,000.