We can buy Republic from the Aztecs as long as we include our horses in the deal, along with 889 gold and 49 gpt, after we temporarily set the sliders to 10-0-0. Losing the use of our horses for twenty turns shouldn't be too painful, when we're fighting a beaten enemy and we have plenty of autoproduced fast units.* Obviously the financial position will become delicate after we return to 7-0-3, at around -20 gpt with 116 gold in the treasury, but I think we'll have various resources to get through the twenty turns: every Russian city we take will reduce our deficit by 8 gpt**; we can hook up our new furs at Samarra and soon the phants at Yakutsk and sell them for whatever they fetch; we can stop most unit production and put cities on wealth (not as weird a choice in CCM as in C3C); and so forth. So I'm confident we can manage the situation and I think we should make the trade.
*As long as we use them carefully and let swordsmen take most of the risks.
**So just by taking Vladivostok, as we're about to do, and Yakutsk, which is close, we'll be almost out of the red.
This plan will require us to be careful about our source of horses at Akkad, which is still close to the front.
Given this, I of course agree with LK and Tusker that we should carry on against the Russians until complete victory. The tech position may be terrible, but as long as we're grinding our neighbours beneath our iron heel, we're winning.
LK, yes, I agree that we wouldn't be revolting immediately, or possibly for some time.