this is not an 'inverter fail' because all of those items you mentioned are actually NOT run thru the inverter, but thru the 12v fuse panel and your house batteries, and while the a/c unit is 120v only, it's THERMOSTAT is also 12v thru the fuse panel, which turns it on/off.
Sounds more like your CONVERTER is acting up, OR you may have a/some loose wires in your fuse panel, OR you have a REMOTE PANEL, such as Firefly, that is causing this situation.
'14 Palazzo 33.3 bunkhouse 34' diesel
KingTailgater2 Dish HD, 100w/5a SOLAR, BlueOx
100,000+mi since '14 - US, Alaska, Canada.... formerly Georgia, then NorthCarolina, now Alabama : )