One thought: I kinda think Charlie knows what Bella is. I'm sure he's heard Billy's stories about the wolves, and when Jacob phases in front of him, it kinda hits it home. However, he doesn't want to know too much, but I think he has some idea that they are vampires.
charlie has a good ideal of what they are, but he is never told. he looks at renneseme and can see how much she resembles the both of them and kind of reads between the lines on that one too.. he just accepts things as they are and is happy for them.