Maybe if they had settled a little further east (e.g on the Yenisei), the Russians would have a bit of a harder time, logistically, invading Sibir.
Alternatively, Kuchum and co. (or an ATL Khan) could have fled east to start their new Khanate, instead of the disastrous decision to go south. There's also Mongolia proper that way (Northern Yuan), so at least they'd be among friends.
Or really, just don't piss off Russia in the first place. It seems like a lot of the blame falls on Kuchum for going fanatic and trying to convert the natives and violently repel foreign influence. Yadigar was the Khan before Kuchum and advocated peace with the Russians, which sparked Kuchum's challenge and eventually victory. It was his subsequent raids that led the Cossacks to retaliate. Maybe if Yadigar remains Khan, things wouldn't be so disastrous for Sibir. They'll probably wind up being vassals to Russia, but still retaining at least some kind of independence and thus survival.