I used to be in the spa / hot tub business, and GFCI's were a constant issue. We always had to use them, either directly in the application or in nearby fixtures. And if they were in the application we were always close to their capacity, were they seem more likely to fault. I have a couple of suggestions. If any of these sound like they're out of your comfort zone I would strongly recommend getting an electrician in to help you out:
1) Make sure you do not have any loads downstream of the GFCI (wired through the back).
2) Make sure you have a good ground in the receptacle. A pigtail to the box isn't good enough, make sure it's solid back to the source.
3) Check your wiring upstream of the GFCI. If it comes directly from the circuit panel check your connections in the panel, they must be tight and secure, nothing loose or crimped. If there is anything else on the same circuit check those. I know it sounds daunting, but a loose connection upstream can give you headaches, too.
4) See if a different load will cause it to fault. Try a heat gun, toaster, something else with about the same current draw. I know you replaced it, but we used to see these run bad in batches. Maybe QC is better now, but 15-20 years ago the receptacle type GFCIs were notorious for false faults.
Only after I did all of that would I consider working on the appliance itself. I feel your pain, these things can be tough. Good luck!