AIPAC was created because it's predecessor had to register.
There's no logical reason AIPAC doesn't have to register, it's just one of those things where a company changes their name and claims they don't know who that old company was.
Citizens United would have made all of this a hell of a lot harder if SCOTUS hadn't ruled the other way. Now basically an unlimited amount of money can flow in from anywhere. It's more complicated than that obviously, but essentially, that's how it is now.