1)Don't run the inverter directly from the van's starting battery. If you don't have any other means of charging, get jumper cables from the van to the house batteries.
2) Connect the inverter to the house batteries with good cable, not alligator clamps.
3)Either buy a bigger/better inverter or remove some loads. A mattress pad or mattress heater is not a "tiny" load. Any time electricity is used to make heat, the load will be considerable. Some transformers do not like MSW power, your laptop brick may be one. Check to see if it's hot.
4)Use a multimeter to see what is happening with your 12V systems. You are no wiser or better off this morning than last night.
5) A freezer or refrigerator has a larger load when starting. My dorm fridge has about a 5-6 Amp load when the compressor starts. A 600W inverter was incapable of running it. A 1000W inverter runs it well.
Good luck with the inverter.