I think Bilbo's offer to take the Ring was also motivated by Pity -- Pity for Frodo, Pity for everyone who has suffered because of the Ring (including Gollum) that he brought out of hiding. Perhaps for him, like Gandalf, the way of the Ring to his heart is by Pity.
I also think that Bilbo has known, or guessed, more about the Ring than we've been assuming. Before he asked Frodo for a peep at the Ring, he says "I heard about the Ring, of course. Gandalf has been here often. Not that he has told me a great deal, he has become closer than ever these last few years. The Dúnadan has told me more." Bilbo was also by Frodo's side -- with Sam -- while he recovered and surely would have learned much from Sam. It's in his reaction to Frodo's revulsion that we can see how much he has discerned: "‘I understand now,’ he said. ‘Put it away! I am sorry: sorry you have come in for this burden; sorry about everything. Don’t adventures ever have an end? I suppose not. Someone else always has to carry on the story. Well, it can’t be helped. I wonder if it’s any good trying to finish my book?'"
Interesting that when he offers to take the Ring to Mordor he again refers to his book.
There is also something in the phrasing of his offer that reminds me of this passage in The Hobbit:
I have got you out of two messes already, which were hardly in the original bargain, so that I am, I think, already owed some reward. But ‘third time pays for all’ as my father used to say, and somehow I don’t think I shall refuse. Perhaps I have begun to trust my luck more than I used to in the old days”—he meant last spring before he left his own house, but it seemed centuries ago—“but anyway I think I will go and have a peep at once and get it over. Now who is coming with me?”
Both times his flippant tone undercuts his serious -- and brave -- offer (and it might be this passage that Gloin remembers with a smile). At the Lonely Mountain he was embracing his Tookishness; I think that at the Council he is trying to make amends for having drawn Frodo into a story that he did not ask to be in.
[Very interesting to wonder what Bilbo and Gandalf might have been talking about -- remembering the first time Bilbo came to Rivendell? Was Bilbo asking Gandalf about Frodo -- if he has well and truly recovered from the morgul wound, if he had ever seemed too attached to the Ring? Had Gandalf ever said, as he said to Frodo, that Bilbo was meant to find the Ring? Or was Bilbo just trying to get more information about this dratted Council?]