I would hook the inverter to the coach batteries. Just an inverter - not an inverter/charger - so just another load. No interaction other than increased load with the charge circuits.
I have not done this mod, but I do have a 400 watt "pure sine wave" inverter connected to my coach batteries. Use it to recharge e-bike batteries, run my wife's "Magic Bullet" food processor, operate my wife's heating pad (bad back), run a soldering iron if needed, etc. Works great.
Not aware of any state not allowing driving with propane on. A couple of tunnels and the Alaska ferry system do require propane off. I found on the Alaska ferry my refrigerator maintained cold quite well for 12+ hours (door not opened, RV on car deck so no sun). I just turn mine off when necessary- for instance, while staying overnight in a friend's driveway where I can't level. Seems to hold cold overnight without issue.
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JLeising
2012 Solera "S"
Calif SF Bay Area
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