You don't need any FAA paperwork to go flying with an instructor. If you are only getting checked out you don't need any TSA stuff either. Show them your log book and tell them your FSDO appointment for the certificate is later that week, they will not care. Under FAA Part 91 regs a flying school can also take you for a sightseeing flight, but only within a 25nm radius, which would cover the bay from Hayward. If you have a log book/UK licence you can just call it an instructional flight.