The fridge doesn't know where its power comes from so if it runs properly on shore power but not on the inverter, and it runs at all (GFCI isn't popped if it does run sometimes) there are really only two explanations that make sense:
1. The inverter can't keep up. It could be under-powered for the fridge. However, I just replaced our Magnum 1000W (with a 4000W) and the new one has a built-in watt meter that says our Furrion is only drawing 240W when it runs. Share the model of your Magnum if you can. We might have the exact same unit, and it was more than enough to run our fridge (but we have different fridges).
A kill-a-watt meter to measure the fridge's draw, and/or a cheap voltmeter to watch the voltage the inverter is putting out, will answer this one quickly. Harbor Freight sells simple multimeters for $5. I firmly believe every RV owner should have one in their toolbox.
2. The batteries aren't as charged as you think they are. You have WAY too much battery capacity for a fridge to kill it. I only have 3x 100Ah batteries. I'm full-time on solar now and running one TV (about an hour a day), a fridge, two laptops, and our furnace (overnight lows are 40 here right now, so it runs, but not constantly) and that draws down my three batteries to about 45% overnight. I have a ton of system monitors so I know they're at last 97% before I go to bed (I don't go to 100% because it shortens the battery life).
This can happen even with no mechanical defect. Lithium ion batteries require special charging settings. And most have internal BMS's now that can override what the charger tries to put into them. If they're misconfigured you might be only at 80-85% by sundown, and depending on what else you're running, this could easily be your issue. Again, a simple voltmeter can answer it. Look up the spec sheets on your batteries and see what they recommend for the various voltage settings, and make sure those are correct in your charger and (if configurable) your BMS.