Suburban SW12DEL........uses 12VDC to trigger the 120VAC to the element
The 12VDC to the On/Off switch goes to DC Relay located in junction box on side of WH Tank
When that relay closes 120VDC goes to the OEM Switch in outside compartment then to/thru AC set of t-stats to element
Inverter is supplying AC Voltage same as Shore Power would
Do you have a EMS --Energy Management System--- that curtains certain AC Loads if too much is being drawn on Shore power
Water heater & Microwave and/or AC Units etc?
That would be only difference between AC from Shore or Inverter
*Inverter not going thru EMS (Energy Management System)
SW12DEL DC and AC wiring Diagrams Page 11
Figures 10 and 11
https://techsupport.pdxrvwholesale.c...ing-Manual.pdf