Balrogs were Maiar, of which Sauron (and Gandalf) were also both examples. We do know that Gandalf, with the limitations imposed on him in his "mortal" form, could have defeated Sauron with the Ring (from Letter 246):
"It would be a delicate balance. ... If Gandalf proved the victor, the result would have been for Sauron the same as the destruction of the Ring. ... But the Ring ... would have been the master in the end."
Given we also know that the Balrog is roughly equivalent to Gandalf in power (after all both died in their confrontation, a nil all draw), it is reasonable to conclude that the Balrog could have defeated Sauron having claimed possession of the Ring.
With that conclusion, the question is now whether it would have. Early versions of the Lord of the Rings (from History of Middle Earth) indicated that this could have been the case:
"'A Balrog!' said Keleborn. 'Not since the Elder Days have I heard that a Balrog was loose upon the world. Some we have thought are perhaps hidden in Mordor [?or] near the Mountain of Fire, but naught has been seen of them since the Great Battle and the fall of Thangorodrim. I doubt much if this Balrog has lain hid in the Misty Mountains - and I fear rather that he was sent by Sauron from Orodruin, the Mountain of Fire.'"
However this was removed from later versions, including the final. As a counter example, we see the Orcs from Moria wanted to avenge their kin on Merry and Pippin, whereas those reporting directly to Sauron had other orders that were not taken as overriding orders by the Orcs from Moria. If Sauron had direct control of the Balrog and by extension Moria, these cross-purposes seem unlikely. All instances of the Balrogs serving with or under Sauron in the Silmarillion are consistent with them obeying Morgoth, not Sauron directly.
Overall I think the Balrog would have claimed the Ring for itself, and then had the potential to overthrow Sauron. There is no canon explanation though, only supposition from the above facts.