Ray, I'm confused. Basically, GFCIs themselves are not problematical and inverters have no personal feelings. When the GFCIs trip it is for a reason.
So, are you saying that circuit the tv is on and the sink is on is powered only by the generator or shore power? And that circuit receives no power from the inverter?
Seems like a simple rewiring issue, as long as the inverter is good enough and your batteries are sufficient and be sure to be able to isolate your alternative power sources from each other. I would be thinking along the lines of solar, better and more batteries, etc.
There are tvs that are powered by 12vdc.
My guess is that if your focus is on dry camping, you really need to get deeper into all the changes you need to make that work better. I don't believe for a moment that the Seneca was designed other than with an rv park in mind, but as Walt said may be an excellent platform to upgrade for dry camping. I think I read where some have added an extra water tank.