It certainly sounds as though the charger portion of the inverter has run amok. What is the fault indication? Are the Leds on the inverter showing overload or overtemp? Or low battery? Do you have a remote display that can provide additional info?
Is the temperature sensor attached to the batteries so the charger can adjust charging to temperature? That can make a modest difference in charge rate.
Also, your inverter can be set for different battery types, e.g. wet cell, AGM or Gel. Make sure the battery type is set to Wet Cell (the type you apparently have, since you say it boils water away). This is covered in the owner manual.
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL