[rickst29]

Long explanation first. The running tow vehicle (SUV or Truck) provides maybe 13.6 volts, but power losses in the wiring will create voltage drop along this path when significant current (and power) begins to be pulled from that power source.

The "+12v" wiring to the fridge comes from the load center. From there, voltage is supplied by either your lithium batteries (at 12.8v) on short and large wires, or the tow vehicle. The fridge wants to pull in about 11-13A (varying in a linear way with voltage available across the resistance coil heater).

The tow vehicle is much longer distance, and contains a lot of smaller wires. Within the vehicle itself, "12v" from the engine compartment might b e using a surprisingly small wire (the original wire in my 2007 toyota 4runner seemed to be smaller than awg-14), and my be about 10 feet long. The actual size of the "Trailer Battery Charge" wire in the cable itself may also be smaller than you expect, that cable comes up into the TM behind the fridge for a total length of perhaps 25 feet and from there, a better wire (AWG-10) runs into the load center.

The grounding return path is "better" in most vehicles, connecting to the vehicle frame after only a short distance of wire from the bargman connector. Within many bargman cables, the grounding wire also also a lot bigger than the TBC wire (I have seen this in a cable which I destroyed by overloading it on the TBC wire.) But the overall path will become "less attractive" than the batteries after the current load exceeds a small value - perhaps just 3-4 Amps out of the 11-13 Amps required by the fridge. Everything else is pulled from the batteries.

The voltage across the load center bus is always constant for al the wires which are attached to it, and your LFP batteries will maintain that voltage at 12.8V until they're less than 20% full. The LFP batteries "hold up" the voltage on the TBC wire, but the TBC wire will not provide more current - if you were "magically" able to pull more current from the Tow vehicle for a moment, it's voltage drop would increase, and your "magical current" would be offset by pulling from the batteries instead.