Yes, bridging the switch, if it’s just a normally open pushbutton kinda thing, is the trick.
And using a relay is entirely safe, as there will be total electric isolation between the Arduino and the switch, so it won’t matter how the switch is connected.
We can further assume, but I suppose with something like a car it would be prudent to know, that the switch is not handling a great deal of current; it shouldn’t have to, so any small relay will do.
For the relay, it is probably most expedient to us a pre-built relay module meant for the Arduino. Google is your friend here.
Powering the Arduino is simple enough also, for testing a cigarette lighter cell phone charger can be used to power the Arduino and the relay board.
Later you can up the mechanical concept on aesthetic terms.
First: light an LED right away, extinguish it after the delay you want. No need to tinker with the switch before you get the software (such as it is) working.