I think that he knew, yeah. The way he looked when Frodo volunteered at the Council was heartbreaking.

And yeah, I agree that Frodo started to get it with Bilbo - I mean, when the guy who sheltered you as a child and who pretty much loved you as kin because he had none turns on you, even for a moment, it registers.

Frodo was very brave and very selfless, doing what he did - willing to give up both life and... wholeness.

*sniffle*