The wire size is important for longer distances due to voltage drop caused by internal resistance of the wire. An analogy is a ¼" water hose out 25 feet and a ⅝" water hose out 25 feet. Obviously the larger hose delivers more volume but at also closer to input water pressure (volts) as compared to the pressure drop in the small diameter hose.
In a perfect world, the wires from the batteries to the inverter should be adequate sized per the National Electric Code for the inrush current imposed be an device with a motor. Resistive loads like a toaster are constant power draw from turn on to turn off.
There are usually only 15 amp rated 120Vac outlets in a trailer, the factory #12 wire is rated 20 amps and is usually on a 20 amp breaker, so that reduces the voltage drop and allows for more devices to run at the same time (up to 20 amps momentary use, but that #12 wire is rated by the NEC for 16 amps continuous load).
We have a 400 amp fuse between our battery bank and the Magnum MS-2812 inverter/charger that is capable of creating 2,800 watts of AC power (can even start and run the rear air conditioner for awhile). The wire size is 2-0 copper or about the diameter of my thumb.
That is why all inverter installation manuals prefer the shortest possible wire runs of larger wire size from the battery bank to the inverter.
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