Will this 555 circuit prevent a car from disconnecting power to the radio when starting?
I don't think this is going to work.

When your engine cranks it's typical to see a large drop in voltage supplied by the battery. Even if you keep the radio from turning off deliberately you're still going to see problems with brownouts. Most circuits do better if they're turned fully off than if they're dealing with this sort of brown-out and I think that's what the wiring in your car is trying to protect you from.

Does the head unit have a manual?