Suggest same as TT. Thameslink or Gatwick Express, then take the little shuttle bus to the CAA. IIRC South Terminal, if coming from train station go to the arrivals door (where the pax come out), past it, then at the far corner down a flight of stairs. Been a while, better check this.
Show them PP, they make copy (happy bureaucrats!), you do same in reverse (happy pilot!). You see - happiness all 'round
Time involved: about 2-3 hours