By the way.
I believe I was wrong about the door jamb switch.
Mine has 3 terminals. That means it's not a simple ground.
Most are a simple ground. One terminal, one wire.
If yours has three, yeah, getting grounded internally might make a difference.
Still, in either kind of dome light, (With and without map lights), it shows a simple circuit from hot through the lamp, to ground.
So that means the ground screw is also for the dome light. As well as the map lights.
I'd still try it with just the one screw to hot and see hat happens.
if the dome light works, and there is no problem... Just for kicks and grins, I would then take out the ground screw. Then move the one screw to the other hot position. Then replace the ground screw and see if it causes the problem.
If it does... I'd just run it without the map lights screw in place. You'd at least have a dome lamp.