A curious idea, I recently wrote down most of my thoughts re: Balin and is ill-fated quest to Moria
here.
But I did not mention the strange comment of Gloin at the council of Elrond that "It was partially in the hopes of finding Thror's ring that Balin went to Moria.
Curious on 2 counts:
1. that he did not know, from Thorin that it had been passed on to Thrain, but not Thorin [goes to show how secretive Dwarves are!] and
2. That Balin was indeed perhaps trying to one-up [or in this case 2 two-up] Dain, by being Lord of Moria
andhopefully the bearer of the Ring of the House of Durin.
In fact that seriously re-inforces the idea of my above thread.
As for Balin being jealous of Bilbo/'s Ring.
I doubt it, but it certainly may have given him the idea that great luck might go with it, just as seemed to go with Bilbo's Ring.