Electrical Issue: Converter, Battery, Something Else?

Update! I ordered and installed a new converter and it seems to have completely solved our problem. Also, it's much quieter and doesn't run nearly as hot, with much less fan noise. Overall, the installation was simple, with most of the difficulty coming from wrangling the wires back into the spaces and getting the new breaker board on.
Thank you all for your advice and help, it's a relief to have a functioning electrical system that's humming along as it should.