My first suggestion would be to find an honest shop.
Even if your batteries were dead that would have NO effect on the AC system.
Disconnect the convertor by removing the it's AC plug. Work your way from the AC source. If you have a GFI that plugs into the AC source plug it in. From this point on anything you plug in causes the GFI to fail you have found the problem. If it holds plug in just the cord to AC. With ALL breakers off in the trailer attach to trailer to the cord. If the GFI holds one by one turn on the breakers, leaving the trailer GFI till last. If all the breakers held till you turned the trailer GFI on the problem may be a faulty trailer GFI of something on that circuit. If everything held to this point disconnect the grund terminals on the batteries and plug in the convertor. if the breaker fails at that point the convertor has a fault.
Now for the batteries. With a voltmeter and while the ground terminals are disconnected test each battery. They should read the Same voltage. Any difference of more than a 1/4 volt could mean the lower battery is drawing down the higher battery. This type of failure would kill them in hours not days when disconnected from AC
Heating elements generally fail open and should not effect the GFI. Disconnect the white wire going to the element and with a voltmeter test between that connection and the disconnected wire while the heater is on AC. If you have voltage across that connection the element is good.
WBCCI 12156 AIR 3144 WACHUNG TAC NJ6
2004 Excursion 4x4
1991 34 ft. Excella +220,000 miles, new laminated flooring, new upholstery, new 3200 lbs axles