What kind of paint are you looking to use? Spray cans, lacquer, enamel, urethane? Catalyzed (automotive two part) paint?
Matte and satin will be pretty similar, every company has a different definition but in my mind satin is just slightly more glossy. Some don't even sell either of those...they will just go flat, semi-gloss, gloss. I see that there are matte and flat clears out there but mostly in spray cans. I wouldn't bother using that for an automotive application... the sun will bake it off pretty quick.
My recommendations depend on how in depth you want to go. Your cheap option is flat Rustoleum or similar paint. It will oxidize and be chalky pretty fast, just the nature of the product.
Second option would be to shoot it with black epoxy primer... will hold up slightly better than Rustoleum because it's catalyzed. Probably pretty cheap, just need to prep it appropriately. Epoxy primer will hold up better than a 2k urethane primer.
Third option would be to prime the vehicle and then shoot it with a 2K (catalyzed two part) single stage automotive paint... many options out there... TCP Global sells Hot Rod Black, Eastwood sells Dead Rat, etc. That's going to be the most durable but also the most work to prep and shoot it.