If it turns out that you have no voltage going into the inverter, then it has to be between the 50 amp (240v) Main panel's inverter breaker and there, since your air conditioners and washing machine, among others, are okay and are breakered out of that box - they don't go through the inverter.
On ours at least, the Transfer Switch feeds the 240v Main. That has breakers that feed the air conditioners, block heater, hydronic heater, fridge (which is why the inverter won't run the fridge dry camping), vacuum, and washer. The last breaker there feeds the inverter, which then feeds the 30 amp (120v) breaker panel that's mounted alongside our 240v Main panel. The things that aren't working on your rig are all fed by that that panel, so you are correct that the problem is inverter related. First easy thing to check is that inverter breaker on the Main panel; if it's good then follow prior posters' leads.
Joel