I don't think it would. The two of them probably were of roughly equal rank in the good old days under Morgoth, so the Balrog would very likely have regarded any attempt on Sauron's side to command it as presumptuous. They might have made an alliance, but the Balrog seems to have been remarkably disinterested in the affairs of the wider world - after all, it spent all those centuries since the Dwarves woke it up doing little more than sitting on its... er... hands in Moria. It's almost like it had decided to retire and only wanted to be left alone. (It only went after the Fellowship after Pippin's foolish stone had disturbed it in its peace in the deeps. Makes me wonder whether our dear Fool of a Took managed to accidentally hit Durin's Bane on the head
...)
On the other hand, maybe not. Why did the Watcher in the Water grab Frodo, of all the Fellowship? Did it just feel some attraction from the Ring, or was it under orders - and if so, whose? Sauron's (who was far away) or the Balrog's (who lived next door, so to speak)? Maybe the Balrog, rather than twiddling its w... er... thumbs for centuries, had been busy all that time recruiting an army of Nameless Things at the roots of the world and was just waiting for the right moment to come out of hiding and Take Over the World - in which case it would have found the Ring quite handy (and what a chance to thumb its nose at its old colleague!). Lots of room for speculation, but little more.
As for Sauron, do we know whether he even was aware of the Balrog?
Finally, I'm not so sure about the Moria orcs serving Sauron in the sense of being under his direct command. Those who took part in the attack on the Fellowship at Parth Galen apparently only followed their own agenda (LotR Book III,
The Uruk-Hai):
Quote:
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'We have come all the way from the Mines to kill, and avenge our folk. I wish to kill, and then go back north.'
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That doesn't sound like they had any special orders concerning rings or halflings, and neither Uglúk nor Grishnákh seems to have thought them very reliable.