Well first off,
You have to stop thinking about component replacement -
You don't know WHAT to replace yet.
Second, ANYTHING that comes in contact with ANY electrical wire (L1, L2, or Neutral) that is WET (to any ground path) will trip the GFCI. Since this was a newer installation of a used tub (at least I thought it was), I would like to think there would be a lot less of what appears to be water leaks or whatever.
Third, There is a pressure switch on the back of the box that has 115 volts going through it... it has a lot of what appears to be white crud and dirt all over it. Id clean that thing up with alcohol or something and be sure there's no moisture anywhere around the electrical switch. Matter of fact, there cannot be any moisture around ANY electrical contact, this includes the back of the pump motor where the wires enter.
Fourth, you still need to disconnect everything like I said above and run the tub for a day or so with no heat, and nothing but the pump. There is no need to remove the wires from the heater element itself, just disconnect from the contactor on the lower right.
For the heater replacement? You don't replace the entire heater, you replace just the element - and at this stage of the game I'd go ahead and replace that ratty looking pressure switch as well.
http://spapartsnet.com/Electric-Heaters ... 6_0_1.html
Select the 5500 watt version.
http://spapartsnet.com/Electric-Heaters ... 0_3_1.html
That's the pressure switch.
That said, don't buy anything until the cause of the trip is found first.
Ummm one more question - are those separate grey and yellow power cables tied to the GFCI circuit?