I will try.
First, you will need an inverter which has enough capacity for the refrigerator load. There are two parts to the load, the starting load and the running load. Then there are two kinds of inverters, pure sine wave and modified sine wave.
A refrigerator may run just fine on a modified sine wave inverter (MSW), but it probably will not be as efficient, and it is hard to tell if the compressor will be harmed by the somewhat strange waveform that they produce. MSW inverters are considerably less expensive than pure sine wave inverters (PSW), although the difference is decreasing. Today, in general, I recommend you use a PSW type of inverter, but you may find the price difference pushes you to the MSW type.
A small residential type refrigerator will probably have an actual running wattage of 150 to 200 watts but the starting wattage will be considerably higher, in the range of 800 to 1000 watts. (I am generalizing here but probably will not be too far off).
So, I would size the inverter to the starting wattage of the refrigerator, that is in the range of 1000 watts.
The inverter should be very close to the batteries you intend for the system, as the wire for a 1000 watt inverter will be large, and properly fused, as it will have to cary in the range of 100 amps at 12 volts for the starting current of the refrigerator compressor. Smaller wire can be used from the inverter to the refrigerator itself.
The running wattage of the refrigerator (assuming a 200 watt compressor draw at 120 volts) fed through the inverter at 12 volts will probably mean the 12 volt
DC system will need to supply between 16 and 20 amps at 12 volts. That is more than most tow vehicle to trailer charging systems can deliver, so your batteries in the trailer will become discharged as you drive, although by just how much is difficult to say because the refrigerator does not run full time.
As to the switching from inverter to grid power, there would be several options, depending on how much money you want to spend on this proposed system. Simple would be switch the refrigerator plug from a grid outlet to an inverter outlet, complex would involve a transfer switch of some kind.
You may find that the somewhat expensive sine wave inverter, wiring, and switching make the residential refrigerator on an inverter a more costly and involved system than you imagine. And if you are not electrically well versed, and are paying someone else to do the job for you, could easily approach the cost of a propane/electric refrigerator. You may have to beef up your batteries in the trailer, as well as the converter/charger to recharge them when you get to your campsite.
This is not a cheap inverter plugged into a cigarette lighter job.
i have outlined my feelings and information on the issue at any rate. Hope I didn't use too many big words....lol.