Do any of the currently produced units have the "AC" in the mold? Molds are not cheap and I would have expected one of the new makers to acquire them when they took over production.
If it uses the sender on their website it is a standard 0.5-4.5 volt analog sender, so where does the 5volt requirement come into it?(rhetorical question).
An interesting puzzle for us but a big PITA for you. I would theorize the 12v supply is dipping when you start and causing a...
You could warm ( don't get carried away) the module and coil with a heat gun to try to duplicate the suspected thermal triggering condition. Hair dryer would be a safer choice, don't make it warmer than you can stand to touch.