YES most rv fridges need 12V DC to operate the controls when using electric or propane.
Most definitely need 12V DC to operate when on propane (controls AND gas valve)
You stated battery is DEAD so the only 12V DC you have is when connected to shore power via your converter (which probably needs a battery to act as a load with it being a 1994 vintage..........plus you really shuld have a good working battery anyway. GET ONE!)
When on shore power
Turn fridge ON (ON/OFF Button)
When in 'Auto' fridge controls will select electric operation.
(Take fridge out of 'auto' and it will probably still run on electric but w/o temp control as long as AC power is available)
So to get it to swap to 'GAS' you will need to leave it in 'Auto' and then go unplug fridge power cord from AC outlet (in outside fridge compartment)
That way you still have DC power via converter (shore power to converter/converter supplying DC) AND fridge controls will see loss of AC power (cord unplugged) and swap to GAS
May take several attempts to get it to fire off .....so if 'Check' Light comes on turn ON/OFF switch OFF then back ON to reset and try again.
ALL of above based on 'generic' operations of late 1990s vintage fridges.
WHAT BRAND/MODEL IS YOUR FRIDGE????
Is it time for your medication or mine?
2007 DODGE 3500 QC SRW 5.9L CTD In-Bed 'quiet gen'
2007 HitchHiker II 32.5 UKTG 2000W Xantex Inverter
US NAVY------USS Decatur DDG31