My guess would be either the batteries have gotten weak or the wiring between the batteries and the inverter create too much voltage drop, could be too small of wires or a poor connection somewhere. When shore power is plugged in the converter is there to bolster the DC voltage so then it works.
First, I would check the battery cells using a hydrometer to make sure they are good and I would clean and tighten all connections.

The 400 watt inverter may not be large enough to start the compressor on the fridge. The inrush amperage or "locked rotor amps" required to start a motor is many times that of the running amperage once the compressor is on.

That combined with the issues Ray mentioned of wire size or battery health might be your problem although 12.9 volts charged is a good sign. How long are the cables between the battery and inverter and what gauge are they?

I assume the red trouble light is on the inverter? What else is the inverter powering when you try starting the fridge?

I use 4 gauge wires about three feet from the batteries to the inverter. I only use one item at a time when using the inverter, usually the TV or DVD and have never had a problem. All connections are tight and all battery cells check out good. The red lights are for the inverter overload. I am guessing this 400 watt inverter is just too small for the start-up as you said. I usually boondock and never use this fridge that is in my outdoor kitchen area but I have a month long trip planned where I stay several days in each campsite that has electric so I thought I would use the fridge and when travelling I would run it off the inverter. Each trip is 4 to 6 hrs apart. Rooftop solar would keep it charged while driving and keep fridge cold. I guess it is a bad idea or get a bigger inverter.

if I plug the RV into shore power the fridge will run while plugged into the inverter.
If the inverter was too small AC wattage wise you wouldn't think it would work even when plugged into shore power either. That's why I thought maybe a problem with the battery or wiring not being able to keep the voltage high enough. But with the aid of the DC output of the converter charger, it can. Makes me think that with a solid DC input it should run. Maybe the battery voltage sags too much and triggers a low voltage shutdown of the inverter.

I wonder if traveling the alternator charging may help its startup.

Good thought, that will be my next project. I will hook up the truck and try it that way. Thanks.

I will end this story here. Either my inverter is just a cheapo that doesn't work well enough, or the mini fridge uses more power than it seems. I tried using it with the truck alternator helping but it blew a 20 amp fuse I had inline. It then blew a 25 and then a thirty amp. That is when I gave up on using the fridge this way. I plugged my TV back in and it still works fine so that is where I will leave it. Thanks for the replies.

Howdy Eddie,
Look at a ohms law conversion table for A/C to DC power to watts. your fridge is running close to 350 Watts. Along with the surge watts on compressor start-up and the power conversion loss from 12v DC to 120v AC in the inverter, you may be hitting the top end of that inverter. You may need to go to a 1000 watt inverter.
Happy Trails!!!

Thank you, Eddie. I believe you are right. I will leave things as they are right now. One day I will get with you and get you to tweak my solar system. Be sure and post if you travel East again, if so, I will find you.

I run a small bar fridge on a 1000 watt inverter. The trick I found is in the sequence of power up. Fridge off, start inverter, plug fridge in with the temp control off. This allows the start capacitors to charge. Turn on fridge to desired temp setting. If you plug the fridge in with the control on, or unplug and then plug in, it overloads the inverter. My unit draws about 70 watts.