Hi, just bought an new toy hauler today and decided to chime in.
Im an industrial electrician, started from resi and commercial. 120v circuits, especially on GFCI has stumped even the most competent electricians. A GFCI senses power leaving the breaker and the power returning on the neutral and if it is off by .8 amps, it will trip.
That being said, the water heater appears to be your problem, simple check, get an ohm meter, any one will do. Disconnect water heater and put your ohm meter where the wires would be landed on the water heater.
You should read something, I'm guessing in the neighborhood of 50 ohms. If you read open, your heating element is open and needs replaced. If it reads 0 or fairly close, you have a short between the element and it has to be replaced.
It may not trip the breaker with no gfci protection because the short circuit is being bleed off to ground, however if you lose your ground for whatever reason, you will have potential running through the water! Meaning you could be electrocuted. Water and electricity don't mix well!
Try ohming the water heater, I have seen many fail open or short. Let me know what you read in ohms. Also look on the water heater nameplate and post that. I can figure out exactly what you should be reading by watts alone, but may include ohms on nameplate. Good luck!