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I see OP is asking about recharging TT battery when running inverter.
The main issue with standard set up is the 30 amp charging circuit, when refrigerators take about 400 amp or +- 40 amp @12v.
Most of trucks today have 120 or more amps alternator, so it is not supply problem, but transfer problem.
Lance makes its own plugs with #8 charging wire in it.
When I don't like the idea about messing with plugs on both sides, I have it on my winter list to run #6 wires with forklift plug to my camper, so I can run it on electricity, while having propane turned off at cylinder.
It is simple safety measure.

I thought the no propane frig models used compressor style refrigerators that were a lot more efficient and ran directly on 12 volts.

bucky wrote:
I'm seeing more units without propane ability for the fridge.
If an inverter is also supplied by the manufacturer to only run the fridge does it stay powered enough from the tow vehicle on long runs to stay cold?

Didn't know they were making RV fridges w/o propane option. That would be a deal breaker for me.

Depends on many things but ... I tried running my RV type fridge on the inverter in my Class A.. Which usually charges faster than a Towed Triler Tow vehicle charges (less wire) and it did not keep up .