Georgia was an ally of the crusader states in the Levant going all the way back to the First Crusade. People in Europe sort of abstractly knew that Georgia existed, but really only that it was a far-away country, east of Byzantium and somewhere around Armenia. They called it “Iberia” because that’s what the ancient Romans called it, and they didn’t know anything else about it (this is of course not the same Iberia where Spain and Portugal are).

In the crusader states there were Georgian monks and nuns in Antioch and Jerusalem, and it was probably through them that the crusaders were able to contact the kingdom of Georgia further north and east. The arrival of the crusaders had weakened the Seljuks who ruled over Armenia and Georgia, and King David IV, apparently with help from 200 crusader knights, defeated the Seljuks at the Battle of Didgori in 1121. Later in the 12th century the Seljuks in Mesopotamia were strong enough to keep the Georgians busy, so Georgia was never able to help the crusaders directly during the Second or Third Crusades.

In 1204 when the Fourth Crusade conquered Constantinople, Georgia supported the creation of a Byzantine breakaway state in Trebizond. It was more or less a tributary state of Georgia, so Georgia’s power extended from the Caspian Sea to the Black Sea and pretty far west along the Anatolian coast. This meant that it was now even easier for the rest of Europe to contact Georgia, through the crusader Latin Empire in Constantinople. They were in contact when preparing for the Fifth Crusade, and Georgia was planning to help attack Egypt, but as you mentioned, the Mongols arrived and defeated the Georgians. The same thing happened again in 1225 and 1239, so the Georgians were busy defending their own territory.

So why did Georgia ally with the crusaders even though they were an Orthodox country? Well you’re right that the Georgian Orthodox Church was in communion with the church in Constantinople, and in fact the crusaders recognized that “they copy the Greek rite in almost all ways.” But Georgia was an autocephalous church, meaning it had its own patriarch and didn’t depend directly on the Greek patriarch on Constantinople. The crusaders also encountered numerous other autocephalous churches - there were also Syrian, Armenian, and Oriental Orthodox Christians, among others. The schism between Rome and Constantinople was a problem for Latins and Greeks but that didn’t really have anything to do with Georgia or any of the other churches. Georgia had no problem with a weakened Byzantine Empire, since that allowed them to increase their own power westward. Georgia and the crusaders also had a common enemy in the Seljuk Turks, so they both felt that an alliance was totally natural and beneficial.

Sources:

Bernard Hamilton, "Latins and Georgians and the Crusader Kingdom,” in Al-Masaq 23 (2011), pp. 117-124

James M. Powell, Anatomy of a Crusade: 1213-1221 (University of Pennsylvania Press, 1986)