Sounds like high resistance connectors to me. To check the inverter, locate the DC output leads. Disconnect the A/C power and take a reading of the DC voltage. Restore the A/C power and take another reading of the DC voltage. Normal A/C off, DC should read in the 12 volt
range. A/C on, DC should read 13.5 volt range.
If you get low DC voltage readings with the A/C on and off, Disconnect the batteries and take the DC readings with the A/C on. If the DC voltage increases, I would have the batteries tested.