I don't see it, really, or at least it isn't easy.
The colonization was done by freebooters hired by minor princes and wealthy merchants who already had some tenuous presense and yasak collection in the North and even across the Urals since the mid-14th c. at latest.
Once the Russians have access to the Urals/Perm and the Chusovaya basin, Sibir will fall, regardless of who they are allied with...unless someone powerful enough can take active interest in preventing the Russians from expanding, and has both the projection ability, the military prowess, and the determination to beat the Russians, and the Tsar is suitably scared and in control of his people.
Crimean Khanate+Nogays+Ottomans seem like a great choice, but the unfortunate thing is, it was tried, over centuries, to no apparent success, in OTL.
Abdul once explored the idea of Astrakhan instead of (rather than in addition to) Vienna. Provided everything goes very smooth for the Ottomans, this time they could beat the Russians and have their Tsar agree to cease expansion of the yasak-paying lands to the East, in favour of the Khanates. Since the Tsars managed to shut down the Mangazeya route when they wanted to, they could maybe do it.
Siberia could not develop rapidly without help from European Russia, at least initially. They all needed food and permanent garrisons to replace and augment the cossacks.
Even so, I cannot see this situation persisting forever. The profit of margin from fur taxes was too great and the opposition too sparse and often ineffective; for Persia or the Ottomans, the reverse is true. They cannot sell the furs easily, and it costs them a lot of effort to support weak allies to no apparent end.