Hm...
Well looking at the full picture here, one must ask if Durin's Bane would even know what the ring of power is. He escaped into Moria before the second age and so would have never heard of the rings of power. So, SHOULD the fellowship have been defeated, and assuming he checked their bodies, it's very possible he would've done nothing with it. Tossed it aside. Maybe he disposed of the corpses by throwing them down the same pit he fell into without closely examining them. This seems a bit more likely to me than him taking the ring and using it. Also I wouldn't say he didn't "help" Sauron, I just highly doubt he even knew what was going on. He wasn't king of the mountain or anything, with lots of servants to do his bidding. He was in hiding, buried as deep in the earth as the mountain would allow. Even the orcs didn't want anything to do with him. I do not doubt for a second that if Sauron knew where he was, he would have approached him for help, and more likely than not DB would've agreed. Obviously there'd be some arrangement, he rules the north, Saruman the south west, and Sauron the east. Something like that.
Of course the catch here is the ring of power has a will of its own. So we must ask would it have some way to "urge" DB to take the ring, or put it on or look at it or whatever. Certainly not impossible. On the other hand, if anyone's power was comparable to Sauron, it would've been a balrog (though Sauron would still be more powerful). So I don't think his will would bend as easily as a mortal man or elf.
Another scenario is a possible connection between two dark maia. What if DB did put the ring on and was able to "alert" Sauron of his presence. Upon Sauron finding him, they (they probably meaning the witch king as ambassador) would make a compromise that the balrog would get lots of power if he returned the ring and helped Sauron defeat the powers of the west.
I don't think it likely a balrog wouldn't be willing to help Sauron. I truly believe that had Sauron known of his existence, he would make any attempt to get him on his side. We have no record of Sauron even wanting the help of a balrog, much less trying to find him, so I would assume he doesn't even know it's an option. Now I don't think a dragon would help him unless LOTS of money and jewels were involved, but a balrog would do it just to cause fire and destruction. Balrogs are genuinely evil, dragons are just greedy bastards.
So thats my take. I don't think he'd know what the ring was. If he did decide to keep it, I don't believe it would wear him down like a mortal man, at least not for a long, long, long, long time, but more likely communicate through him to bring Sauron in. Honestly it seems more likely an orc would find it before the balrog. Once the fellowship was done, DB would go back into hiding. He's not interested in loot....