It all seems normal except that freezer, which as said above, may not like MSW.
You get the inverter alarm when battery voltage input is down to 11v (loaded voltage) The van will idle at about 14v with the inverter running loads. It helps to turn off the van's climate control so more juice goes to the inverter job. it ought to work within limits.
You have two limits-
1. The inverter's own limit . Your loads are acting normally where you can adjust the amount of load, but you are seeing battery draw down when too much load is applied, because of the second limit
2. The DC source limit. The van can't maintain the inverter once the demand gets to a certain amount even if that is less than what the inverter could do with a better input.
When I ran an inverter off the truck battery to run a battery charger from the inverter, I found that the truck's voltage would hold at 14v (watching the dash gauge) up to a point, but if I added more load to the inverter, the truck's voltage fell off rapidly.
This is what is happening with your van. With an inverter load the truck could maintain at 14v (truck voltage) I was able to idle for an hour charging the trailer battery, and I could have kept on longer. There were no issues (truck would run out of gas eventually of course)
I also had trouble with the inverter being too close to the engine. the inverter shut down from the heat. I had to move the inverter farther away and that solved that.
The freezer load should run ok, so that must be a MSW vs PSW issue I suppose