What can happen with a SCOTUS finding that the Constitution was violated by the Jersey Supreme Idiots is that should the Pubbies grow a backbone upon getting a majority, they could refuse to honor a Lautenberg election on the basis that the election was un-Constitutional. At that point, I honestly can't answer what would happen.
If they decide that the election itself is unconstitutional, then I'd think they'd have to throw out all the results and start over. Otherwise, it could set a really nasty precedent. A candidate who loses an election could then object on the grounds of any type of technicality that the election process favored his competitor, and he - the person that the majority of the people voted against - would be ushered right in.