Hi Jasper,
Yep, unfortunately the F lounge is closed at night. The Airport Novotel is within walking distance and TG wont take you there but the hotel has a minibus service to get you there. It's a decent hotel but definitely not cheap for a short stay.
You'll have F lounge access on your return flight and should be picked up by a buggy. Not sure if you'd be buggied to your connecting flight on a different airline, we've only gone TG to TG. Make sure you do the spa! I thought I wouldn't be into it but I'm hooked now.