The installation and operation manual seem to say the 12 volt gas system is not connected to the 120 volt electric heat system. I do not see a relay in the 12 volt diagram.
According to an etrailer.com support person, Dometic GC10A-4E SP Water Heater and Atwood AT94022 are the same model under new Dometic management. www.etrailer.com
Operating instructions indicate electric heating circuits are separate from gas heating system. Gas heating system uses 12 volts. Electric heating uses only 120 volts.
Instructions for electric heating do not reference the remote lighted switch required for gas operation. See page 4 of instructions.
Most RV water heaters I worked with use 12 volts for both gas and electric heating. This model apparently does not.
I expect the blown 12 volt fuse labeled “pump” does not affect electric heating. If it is indeed the pump, it won’t affect gas heating as well. If it is miss-labeled and includes water heater 12 volts, then it likely only affects gas heating.
Reset electric heat:
“If the manual reset high temperature limit switch should operate, reset switch (FIG 11-A) by depressing with pencil or other nonmetallic object. If still experiencing problems contact an authorized Dometic Service Center or Dometic Service Department.”
http:/www.dometic.com for service issues.
If the reset device and or the "on/off" switch is broken, that may be why electric heat does not work.