"Here Bhagvan clearly says worshiping other gods is wrong or is an unauthorized manner."
Krishna doesn't say that. He just says that the method is not proper, but the worship does pay its due by reaching to him indirectly. Here is the correct translation:
BG 9.23 — The devotees of other DevatAs worshiping (them) full of faith, they also worship "Me" only with non-methodical way.
This "Me" creates a great confusion and also gives a chance for bad criticism.
Krishna's "Me" doesn't refer to his personal form of "VAsudeva" alone. During discourse of Gita, Krisha is believed to be in the highest spiritual level of everything, which we call as BrahM (or Brahman). This is the 4th state in every aspect. Different names, but same meaning. i.e.
- Among modes, it's Atma/ShunyatA (beyond, Sattva, Rajas, Tamas)
- Among awareness, it's Turiya (beyond, JAgrut, Swapna, NidrA)
- Among enjoyers/Purusha, it's Moksha (beyond Dharma, Artha, KAma)
- Among visuals, it's Divya Drushti (beyond height, width, depth)
- Among acts, it's Leela (beyond Dharma, Adaharma, Vidharma)
- Among Karma, it's Akarma (beyond Satkarma, Dushkarma, Vikarma)
The "VAsudeva" (Krishna as person) is also "worshipable" (BG 10.37), but so is "Arjuna", "PrahalAda", "VyAsa", "RAma" and many more described in Gita chapter 10.
"So is worship of Prabhu Shriram is wrong or Unauthorised?"
A devotee gets compelled by own self, for whom to worship. In Gita, RAma is anyway included:
BG 10.31 — Among purifiers, I am wind; Among carriers of weapons, I am RAma; Among aquatics, I am the crocodile; Among water streams, I am JAhnavi (Ganga).
RAma can be Raghupati RAma, Parashu RAma, Valadev Bala RAma. Similarly, "Wind" can be Pavana-putra HanumAna, VAyu-putra Bhima. In spirituality, take everything as perceptions or choices. Being judgmental, may create conflicts.
But what if someone 'worshipable' hasn't "made" into the list of chapter-10? :-) Luckily, Krishna was aware of this. There can be many entities pre & post Gita, which were quite worshipable by various beings. Hence Krishna ends the list with following:
BG 10.41 — Whichever influential excellence(Sattva) [entity] certainly has prosperity(Shrimad) or energy(Urjita); those you know to be possible as part of My aura only.
We can even worship our own self (BG 10.20) as well; because "Self"(Atma) is actually Krishna's "Me" only. Similar to Advaita or Buddhism's ShunyatA.
"Is chanting name of Lord Shriram is unauthorised worship of almighty god although he was one of the avataras of Lord Vishnu?"
"Chanting(Japa)" anything with devotion is fine. Be it, "RAma, RAma, RAma" or "Krishna, Krishna, Krishna" or "Hare RAma ..." or "ShivAy ..." or "Aum ..." or GAyatri or even "Money, Money, Money" (Kuber BG 10.23).
BG 10.25 — Among great Rishis, I am Bhrugu; Among voices, I am the indestructible (Aum); Among perseverance(Yagna), I am continuation of "Chanting(Japa)"; Among stationary-s, I am the snowy place(HimAlaya).
Importance of Chanting
Chanting is one methodical way to fix one's consciousness to the 'entity' being chanted.
Throughout the life [un]knowingly, we are simply preparing our "state of consciousness" for death. During the final moments, whatever our consciousness remembers or bewilders or explores, that is achieved by it. If the part of it is fixed on supreme and part of it is in worldly matters, then the former part disappears in supreme and the latter part returns back to the worldly matters.
[BG 8.6] — Also [when] the body is left remembering this or that "Nature" at the end, O son of Kunti, [it] always approaches towards establishing those nature(s) only.
"If one worships Bhagavan Shriramkrishna then will it be unauthorised?"
No. It's quite fine. He was after all a BrahmachAri. Discard the notion of BrahamachAri as "person not having sex". Rather it means, the person who "acts as per god's will" (Brham + AchAra). Whatever they do, is one or the other form of leela. During death, if a person's mind is fixed upon Shri RAmaKrishna Paramahamsa, then his/her consciousness will get whatever state of him only.
In fact, I would go on to say that, if our heart suggests us to worship something so called "sinful" or "unauthorized", then we should do that. Act of "Worship" fixes our consciousness towards one-ness. Ultimately that one-ness will merge with other one-ness and finally the supreme one.
In Jainism, RAvana (& Karna) are eligible to be Tirthankara along with Sri Krishna, BalarAma, VyAsa. Because, whoever has fixed their mind unto supreme (with any kind of emotion), tends to become the same.
BG 9.30 — Even if an evil doer worships me with no other devotion, is considered saint(SAdhu) only, for being situated rightly.
Already seeing good answers & comments. Since you asked to answer, I am giving another point of view, to support that worshiping other forms is correct.