GN is an official status granted in some states based on your having applied for licensure in that state, not just being someone who graduated from nursing school. AFAIK, you would have to have applied for licensure in your home state (which is what you're wanting to avoid) in order to be granted GN status (if your home state is even one of the states that still offers GN status; fewer and fewer of them do as time goes on).
I agree it's going to be hard to find a nursing job just for the summer. If you find a job without telling them you're planning to leave by Fall, you will leave that employer on bad terms, and that could come back to bite you in your job search
in the new state (or at any point in your future, for that matter). Could you just focus for the summer on passing NCLEX ASAP and job-hunting in the state to which you're planning to move? That may take longer than you expect and, the sooner you get started, the better. Can you move to the new state sooner, instead of hanging out at home for the summer?