I don't think Bilbo's motivations to take the Arkenstone and sneak it out to Bard was driven by the Ring. The Ring works by delusions of supreme power, by possessing it and using it. Boromir wants to take the Ring and use it to command armies against Sauron. Sam's temptation is to use it's power against Sauron and grow an extravagant garden in Mordor. Gollum's temptation is to keep it for himself and dine on the finest fish forever. The Ring works by tempting the bearer with a delusion of "claim ownership of me, you can overthrow Sauron and use me to achieve your heart's desires."
Conceptually, the ring in The Hobbit isn't the One Ring in The Lord of the Rings. But even if it was Bilbo's actions to take the Arkenstone were not driven by The Ring. The Ring would not be tempting Bilbo to
use the Arkenstone to achieve his desires of reconciliation. It would be tempting Bilbo to
use the Ring itself, as the means to achieve his goal. That's how the Ring works to gain a grip on it's bearer.
The original topic has certainly been interesting to ponder...and drag this wight out of his barrow
. I haven't read The Hobbit in quite awhile, but it seems to me Bilbo felt forced to take drastic and, in the very least questionable, action in an attempt to be a peacemaker. The result doesn't go as Bilbo planned, but the result doesn't change his intentions. Something you can see in Tolkien is good intentions may lead to unintentional consequences, just as evil intentions may lead to an unexpected positive result. Whatever side you come down on, if you think Bilbo's sneaking and concealing the truth was morally questionable/treacherous (or that he had a legal claim and intentions to bring peace), he's later absolved by Gandalf and then by the person he committed the offense against, Thorin.
Bilbo is by no means perfect, this reminds me of how he concealed the true story of how he got the Ring for many years. However, Bilbo concealing the full truth is so very minor, it's silly to condemn him, or act as if he's committed some great injustice. When confronted with the fact Bilbo hasn't been completely honest, he comes clean, and that's more telling of Bilbo's character than trying to conceal the truth because he's too embarrassed at the time.