Inverter Bad?

You need a lot more diagnostics before you can determine where then problem is. It could be the inverter, batteries, converter, microwave, a relay, fuse, breaker, lose wire and probably a half a dozen other things. You need to start checking inputs and outputs on all the pertinent devices before you start throwing your money down a rabbit hole. One thing I can tell you, starting the engine of the RV with the battery bank at 10.5 Volts is more than likely not going to be sufficient to run the microwave. Secondly, it is highly unlikely that the stock systems in a 20 yr old RV will ever run the microwave without serious after market improvements. Most importantly, lacking a full and complete description of your whole system and I/O’s, no one on here could reliably diagnose your problem(s).

 what type are the house batteries, and how many? it almost sounds like the batteries are done, did you run the batteries dead at all?  it could be the inverter though.  when you say the volts drop to 10.5 to 11 V is that measuring at the actual batteries?  there shouldnt be that much of a drop if the batteries are good.  this one will take some trouble shooting.