Personally, I think you've got this all wrong. Don't stay at the airport, but in London. Arrive, go straight to the hotel in London, then go out to dinner. 9ish is a good time for dinner on a Saturday night and choose a fun spot. Also, if you are heading TATL on Sunday, you need to start adjusting your time, so keep it late.
Then get up late-ish on Sunday and head to the airport. For a 12:30 flight with a 10:30 check in you'll have to leave at around 9:30 so there isn't much time to do anything before then anyway.