I often think the Balrog must have found plenty to keep itself busy in the deep and dark places under Moria - maybe tormenting the nameless things that Gandalf says live down there. The Balrog might have been like the Nidhoggr, gnawing at the very foundations of Middle-earth
Or else he/she/it chose to keep out of Sauron's way. It may not have wanted to be used and exploited by him like so many other beings and creatures were used by both Morgoth and Sauron.