We have different equipment, Xantrex Freedom 3000w inverter and a different remote, but when we see this happen, I first try going through the settings to see if the "invert" setting has changed from INVERT to STANDBY, or some such terminology. In the past, I have cycled the setting through STANDBY and back to INVERT and was successful in making the INVERT light switch back on.
You're getting 120v when plugged in because the 120v is passing through the inverter and, therefore, the inverter is not needed to convert 12v from the batteries to 120v for your 120v circuits. I am hoping your inverter is simply confused and needs a helping hand to get started inverting again.