They'd still have to repel the Turkmen invasions of Aq Qoyunlu/Qara Qoyunlu. But if they aren't as devastated by Timur's campaign, they'd probably stand a better chance at avoiding the damage that caused the fragmentation of the Georgian state, but I'd expect the victories to be costly and take a generation or two to recover from, and at that point Georgia is torn between the Ottomans and Safavids, just like OTL. And given you have the Golden Horde surviving, that adds another potential threat.
Timur more or less destroyed Christianity in the Caucasus outside of Alania/Ossetia, and probably permanently destroyed the government institutions (the Mongols had already done a lot of damage) that led to the familiar tribal nature of Chechnya. But Christianity was never deeply rooted (outside of Alania), and even before Timur, the Lakhs and some nearby nations had already converted to Islam. Being surrounded by Muslim nations gives a real incentive for the peoples of the region to convert. Conversion was mainly the work of Sufi orders, so the Georgians would need successful missionaries and churchmen in these reasons to keep the incipient cultural sphere they had together.
