"Jumping power directly to the fuel pump causes the fuel pump to run, allowing the car to start and stay running"

Are you sure the Relay is good ? This is just a thought, but from reading your post, it almost sounds as if the relay might not be good. Can you try a different ( borrow..)FPR in it's place to see if the problem goes away? As you know the FPR issues can be either intermittent or plain difficult to nail down because of miniscule solder cracks.
Good luck with it.