I don´t think the Qing will be able to stay for a lot of time in Burma and I think they will probably only take part of it(the coast would go to Britain) and lose it in an alternate version of the series of rebellion it experienced.
I think that they would probably only take the northern areas of Burma, which are the least Burmese parts of Burma (Shan state, for example). Any further would be very difficult to maintain, but suppose they somehow did it.
I can see the emperor ordering a mass extinction of the Burmese people who keep rising up in a similar fashion to the Dzungars, and then resettling the area with Han, Hui, Manchus, and maybe some Tai.
Because of the POD, I think that Britain and China might clash over Burma earlier than 1836, which leads to a stalemate (or pseudo-Chinese victory, since it's not possible for China to invade Britain - oh wait, British India, but that would be much harder - but it'd also be extremely hard for Britain to invade China because Chinese junks are more useful against British ships of the time period).
Unfortunately, this may delay modernization efforts (or not at all, given that the war is earlier, and there's time for another war later).
The Russians would definitely back up the Qing much more than ITTL, since if they can get a modernized-ish Chinese army, they have a HUGE bargaining chip over Britain. Literally, this Chinese army has easier access to British India, and while it would be harder for the Russian army to reach British India, this means that China becomes a part of the Great Game.