1. For a purpose of running a residential fridge, choose one that has a built in transfer switch. When there is AC shore power available, it will choose that. When you unplug, it will draw from the battery. No muss, no fuss. This is an illustration, not a recommendation:
https://smile.amazon.com/APT1200-12...fix=inverter+with+trans,automotive,175&sr=1-5
2. When installing the inverter, locate it as close as possible to the battery, and use HUGE cable to connect directly to the battery. Add a fuse or DC breaker in the line unless the inverter has its own input protection circuitry.
3. For the transfer switch function to work, you'll have to supply the inverter with a 120 volt feed...best option is straight from an open breaker on the panel, but you could pick-off power from a nearby outlet...assuming the chosen circuit has the capacity to support the intended load. Another way to approach this is to snag the wire feeding a nearby outlet, use it to feed the inverter, then plug the RV outlet into one of the outlets on the inverter. But be aware that all circuits in an RV are likely to be fed through a GFCI, so tripping the GFCI would shut off 120 volt power to the inverter and transfer the load to the battery. You might not notice this, but with those loads, if you're connected to shore power, the cooling fan on the RV converter/charger would run continuously as the battery is pulling a 50 amp charge through the converter. Details.
The most important thing to remember is that the inverter supplies 120 volt power from a 12 volt source, so ALL calculations relating to load must be based on 12 volts. Let's say you install a 1000 watt inverter. That inverter will draw 100 amps (that's right, one hundred) when supplying 1000 watts of 120 volt power to your load. Your fridge may be more like 500 watts, so the load from 12 volts will draw 50 amps of current from your battery bank.
Also add about 5% to 10% for inefficiencies in the inverter...and the power it consumes.
There's a chart on wire size at 12 volts for any given distance here:
12 Volt Wiring: Wire Gauge to Amps | Offroaders.com
But most installers would use 2/0 AWG wire to connect the inverter to the battery bank. This "welding cable" minimizes losses at any load and preserves your battery bank. Smaller wire will waste power in resistance that becomes heat.
Note that I said battery bank. A typical single group 24 battery installed in a small camper can only supply 35 to 40 amp-hours before needing to be charged. The "gold standard" of 4 x 6 volt golf cart batteries wired in series/parallel can deliver about 225 or so amp hours. If you're fancy and rich, Battleborn and other LiFePO4 batteries can do even better, but at a kilobuck/battery, emphasis on rich.
Anyway, to run your fridge and the inevitable other 120 volt loads you will be tempted to add, you'll need a big battery bank.
For argument's sake, let's say your fridge draws 600 watts to start, and uses 300 watts to run. Again, for argument's sake, lets assume you're camping in TX in the summer, and the fridge is running on a 50% duty cycle....this makes the math easy. 300 watts at 12 volts is 30 amps. 50% duty cycle means running for 12 hours. 30 amps/hours x 12 hours = 360 amp/hours.
Ooops!
Your real numbers will differ and probably be quite a bit less, but here's the calculator to help you figure things out:
https://www.rapidtables.com/calc/electric/watt-volt-amp-calculator.html
Remember, EVERYTHING through an inverter is 12 volts, even if the item is a 120 volt appliance. The ultimate source of power is your nominally 12-volts battery bank.
Unless you like the drone of a generator, you'd best investigate solar if you plan to boondock with a residential fridge. That's another whole big topic.