If by the “Crimeans” you mean the Crimean Khanate, it became an independent state only in 1441 and before this it was a part of the GH so the attempts to kick it out earlier would be extremely unhealthy. Actually, Kaffa (Feodosia) had been
purchased by Genoa from the GH in the late 13th century and prior to this it was just a minor settlement with a mixed population. Having the Genoese trade posts on the coast was convenient to
both sides and Kaffa became one of Europe’s greatest slave markets.
A rather fantastic scenario (“kicking” the Tatars out of their ulus located so close to the center of the GH would be quite a military fit) that you mentioned implies an economic destruction of the “Republic of Kaffa”: it existed on the transit trade and having GH as an enemy would mean cessation of the trade.
Ability of a state to survive
economically exclusively within the peninsula (Crimean Khanate in OTL had a much greater territory) is an interesting question but the answer is probably negative. In OTL the CH was producing some export items besides the slaves (knives, carpets, salt, etc.) but, even with the reasonably well developed garden-based agriculture, it could not feed itself and relied upon the grain supplies from the Ottomans. Without money from the slave trade there are no funds available. The “whole peninsula” looks good on a map but it’s central part is rather dry and not very hospitable: you may notice that most of the cities are in a coastal area. So to change the OTL agriculture into something self-sufficient would be a formidable task during the Middle Ages.
Ability of a state to survive
militarily is even more problematic. To start with, the GH would be trying to crush the rebellion and defending the isthmus of Perekop with the forces realistically available in the Peninsula is not realistic, especially keeping in mind possibility of invasion through the Kerch Straight and a relative ease with which the peninsula can be accessed through the Syvash.
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And if by whatever reason the GH fails to do this, here come the Ottomans with their fleet, superb infantry and powerful artillery.