I would try a gentle leader before a prong collar, their necks are way to sensitive and fragile for a prong collar. In fact I would NEVER use a prong collar on a basenji (or any dog IMO)
As for the car, I don't know what type of car you have but I would suggest a crate if he is that riled up. If you can't put a crate in your car then best way is to desensitize him from you getting in and out of the car, you can do this by just sitting in your drive way, and getting in and out of the car with him, teach him the wait command so that he has to wait until you are ready for him to get out. And if he doesn't do anything while sitting just in the drive way, drive to the first stop sign, turn around and go back home and repeat the process of getting in and out, your neighbors may thing you're crazy, LOL, and it may take time, but have patients and stay calm. If you can use a crate I would still work on desensitizing him of getting in and out of the car and what is appropriate behavior. Use lots of praise and lots of treats.
As for the seat belt, keep it on him when you are in the car, when you're not take it off him but leave his leash on him. If he is a leash chewer also try dosing it with Bitter Apple or something like that.
Here are links to the gentle leader and bitter apple.
Bitter Apple: http://www.1800petmeds.com/pdetail.asp?SK=10575
Gentle Leader: http://www.premier.com/pages.cfm?ID=29
Also want to add, the gentle leader is NEVER to be left on when not supervised, and there are many dogs that it can take a long time for them to learn to get used to it and tolerate it (some adjust quickly). And never is to be used as a form of punishment. Okay sorry I'm off my soap box now on that.
Hope this all helps and can lead to a happier human and a not as naughty basenji