For the refrig, the heater element relay may be damaged. Here is a website for parts and many of the service manuals to help you troubleshoot. Perhaps the tests will lead you to the part needing to be replaced. https://myrvworks.com/manuals/
Edit: Oh dang you are in Seattle, you may just get hold of the folks who run the website, they are in Port Angeles surely they can help!
I would probably replace the Water Heater controller board at this point but you can continue to troubleshoot. The same website has water heater manuals also.
Inverters are expensive. I would try to track down its service manual also or send it out for repair.