Well, I had been grounding the inverter to the negative battery terminal, so I switched to grounding it to the trailer tongue of the camper. It didn't make a difference. So, I went to Harbor Freight and bought their 2000/4000 watt inverter, came home and hooked it up exactly the same way as the other one. AND IT WORKED! PERFECTLY! I can even run my microwave! The other inverter is going back to Amazon tomorrow.
So it wasn't anything in the way I was connecting everything, it was just that inverter.
Thanks everyone for your helpful input. I have certainly learned a lot about inverters in the past few days.