Personally I would be fairly comfortable taking the 05.33 from Blackfriars, arriving Gatwick at 06.18, for a 07.30 flight with HBO. Even if that train is cancelled, there are two later backup options which should still get you there in time for your flight.

For even greater comfort you could take the 05.05 which is plenty safe.

That said, if you are happy staying at the airport or taking a later flight, those may prove less stressful. Only consider staying at the Hilton, Bloc or Hampton (North terminal) though - no point faffing with the off-site hotels.

You can buy Fast Track on the LGW website I think.