I think this is a fair question for debate. Your summary of the 22nd Amendment is correct, but the 12th Amendment is pretty expansive, stating that no person who is constitutionally ineligible for the office of President shall be eligible for Vice President.
I don't think this is an easy question at all. Obviously, B. Clinton is no longer constitutionally eligible to run for President, so is the 12th Amendment triggered? But as you pointed out, the 22nd Amendment just states "elected." This seems to be a careless drafting in my opinion, but it states "elected" nonetheless.
Presidential succession is a tricky thing. I'm not even sure that the Supreme Court would address this issue. I think it might simply sidestep it and call it a political question.
But, Bill Clinton is NOT Constitutionally ineligible for the office of President. That phrase (probably) does not encompass the limitations of the 22nd, but even if it did, the ineligibility described is a procedural one that would not be triggered if he were not elected.