I'd bet the second bulb isn't even wired, it just came with the stock fixture (Backup lights). The first bulb is the key, with one filament being for the running lights and the other for the brakes
and signal lights, which was standard automotive wiring until recently.
Do you have yellow signal lights on your tow vehicle? If so, there needs to be a diode block or relay set in the TV's wiring to prevent voltage to the egg's dual-purpose filament from coming back to the wrong fixture on the TV (dash lights lighting
might be a symptom of this). Diode blocks available at Wally and auto parts places.