My previous trailer's fridge was 120V/propane and we didn't want to run the trailer with the propane on; propane was great for off grid stationary & fairly level.
At least that's what the RV vendor (Keystone) drilled into our heads.
I just had a quick talk with an Oliver sales rep and he confirmed that the fridge is a 120VAC/12VDC unit and you can run it while towing.
As for charging the batteries from the tow vehicle, he wasn't sure if the batteries could be charged via the 7 pin, what the current draw would be or if it's even connected.
So the 12V aux pin on the LC is fused at 40A and energized via relay.
I'm assuming (uh oh

) that it must be good for something...30A give or take to the trailer while under tow would be great!
Per your suggestion, Oliver sells a solar upgrade with solar panels on the roof which looks good at first glance.
The sales rep said, with that option, the solar panels would indeed charge the batteries while under tow, but of course you need sun LOL!
Anyway, the Oliver rep. said he'd get back to me with the particulars on the 7 pin.