1. Go to Ethnologue and look at tibeto-burman langs listed there.
2. Ethnologue usefully has a list of languages sorted by language families that includes subbranches and lists what country the language is spoken in here
3. With the information given here, I can therefore ignore everything not in india, as well as the Tani and Kuki-Chin-Naga subbranches. There's of course still a shitton, but I open each language in a new tab.
4. I now go through every tab and check for number of speakers. Or, well, I would, but ethnologue is not happy with me:
That makes things a bit more annoying.
5. I close all those tabs, Firefox informs me it was 56 of them. I go back to the original list I linked and open a second browser window. Another thing ethnologue lists is ISO 639-3 codes, which one can also search on Wikipedia, so I do just that. 56 times.
6. Wikipedia luckily has articles on every single one of those 56 languages, complete with a number of speakers sourced to Ethnologue. I can narrow it down to five languages with "around 10k speakers": Ranglong (8k), Pattani (11k + 5k L2), Takpa (12.4k, 9.1k of which in India), Zangskari (12k) and Atong (10k). Curiously, a very large amount of languages seem to have 30k or 65k speakers.
7. Assuming one of these five is the language we're looking for, I take a look at the wiki pages first to see if any of them obviously fall away:
-Atong does seem to be written, but mostly by simply writing in Garo or using its conventions. This seems to fit your description of "Not of it's own" quite well
-Zangskari is written with the tibetan script, so I can probably discard it
-Takpa appears to also be written with the tibetan script
-Pattani appears to be unwritten
-I can't find a whole lot on Ranglong but it's speculated to be Kukish, i.e. Kuki-Chin-Naga, so I can probably exclude it for now too
Based on these informations, I think Atong might be the best guess.