3 amp output on the inverter would mean about 30 - 40 amp input to the inverter.
You have to make sure it is a pure sign wave inverter to start up a high torque motor, such as used in a refrigerator. Not a "Pure square wave" or other trick wording to sell a modified sign wave inverter for much more money.
I would suggest a inverter with 600 - 1000 watts capacity, and maybe 1,500 watt or 1800 watts such as the Xantrex 1800 watt inverter, so you could run the microwave as well.
As for batteries, I would suggest a pair of batteries. Deep cycle will last longer, and a pair of golf cart batteries at 67 pounds each will store 2,000 watts between the pair, so you could run the inverter and still have some power left in the battery 3 hours later.
The inverter will have a 75 amp surge input while starting the compressor. So #4 wire is justified, and 100 amp fuse.
Your 120 - 130 amp alternator should work fine. I would suggest a new isolation relay between the alternator and the battery, such as Grainger.com 6C017 that is rated at 125 amps and has silver allow contacts. Have them energize when there is 12 volts to the run position on the engine. I wired mine to the +12 going to my radio.
Personally I would not suggest wiring the inverter to the RV electrical system. You can plug in the refrigerator directly to a cord that is coming from the inverter, and place the inverter as close as possible to the battery, but not in the same compartment with it's acid fumes. Then just plug the refrigerator or TV and laptop to the receptacle powered by the inverter, or plug it into the RV - as needed.
Good luck,
Fred.