Heathwick. High-speed rail link between Heathrow & Gatwick - if it's as fast as HS1, or the St Pancras-Stratford Javelin it'll only be a 15 minute journey. You don't even need to "check out" as it'll be one big airport with the platforms at both ends within customs etc. Get the most out of Gatwick's capacity without taking all the business away from Heathrow.
Heathwick
Heathwick is an absolute non-starter. Passengers are not going to transit through a hub airport which is actually two airports 30 miles apart however quick the transfer, when there are plenty of much better and quicker options available (Amsterdam, for example).