jack totally knows. in the first film he and barbossa went off on hats a lot. Jack KNOWS what Barbossa's hat looks like. IMO. and when he picks it up, he is seriously studying it.
Unless he just think Tia is collecting dead people's things or something. LOL. But Jack is really sharp. I bet he at least has some suspicions. I mean, if Jack and Tia really go way back, he probably knows she resurrects people as a hobby, and he might even guess she will resurrect him after he jumps into the kraken.
I agree that Barbossa was definatly resurrected. I did not notice that the hat and rings belonged to him, although, when Barbossa walks down the stairs look closely at his teeth. They are silvery and his gums are green. Just like Tia's. But, there is no idea of telling what will happen in the third movie!