Hmmm...new coach is under warranty...right? Maybe it's time for a trip to a qualified repair facility. I know, you're probly like me and would just as soon fix it yourself, right?
Here are my thoughts on your issues based on my experience:
1. Does the garage gfi trip on anything else? Seems to me a gfi senses a short and trips, so that tells me there may be something up with your wiring or solonoid. I can't say what exactly though.
2. Batteries are supposed to bubble but not a lot and not for a long time. If they got run dead between the factory and the dealer and then charged up and delivered to you it is possible they have been ruined and need to be replaced. I would remove them from the coach and charge them seperately, or have them charged and load tested by a qualified repair facility. You can also do this yourself with a proper charger and hydrometer. All of the cells should match. It is quite possible you have one or more bad cells in one of the batteries and that's why the bank won't fully charge.
3. Same issue as #2.
4. Charge voltage for 12v deep cycles is 14.4 volts. For 6v golf-cart batts it is 14.8 (per Trojan). So it is entirely possible to have that kind of voltage reading soon after removing the charge source. A 'normal' full charge voltage of 12.7 to 12.8 should show after resting overnight. If there's a dead cell that battery will discharge the whole bank as you describe. That's why pulling them out and seperating them will help you troubleshoot.
5. It certainly sounds to me like this item is related to #1. That's why it trips the gfi. I'm guessing there's an issue with your inverter. May even be causing the battery issue. You have to isolate everything so you can properly test.
Not enough information to ascertain if all issues are related, but it does begin to seem that way. When you turn on your inverter and it won't operate that's pretty clear there's a wiring issue or failed component. I'm betting on the latter, likely the inverter. If it's shorted it would cause the gfi trip and drain the batteries. I would isolate the inverter out of the picture and see if that fixes one or more issues. And certainly check out the batteries as well.
Good luck. Can't wait to hear how this shakes out.