For a budget car challenge, you could "buy" cars under a certain value and then do several tests.
So start off, steal a car, make sure it's sell value at the customs shop is under... say 2k or 1k. Then some challenges and races: drag, road, and off road,
For off road I was thinking the bike path to the top of Chilliad, or one of the ultimate adventure hill climbs. Of course there is always the opposite and going down Chilliad the hard/fun way.
Amphibious test to see who's car can go the deepest into the water and back before quitting or a river crossing/path. IF it dies you're out... or get the "punishment" vehicle, a panto or whatever.
The customization challenge. Make a car look the most fitting with in a theme. like most British, most gangster, most trashy, most beat up. Most beat up could be actually using a bat or other weapons on your car.
Some random more GTA than Top Gear inspired ideas for budget cars:
Ramp test to see which car flies the farthest.
Bowling... Which car takes out the most players in one or two passes.
Survival... with a 4 or 5 star wanted level last the longest with out getting wasted. This one could be as teams, car loads, of people.
In keeping with the spirit of Top Gear, aside from the customization challenge, NO car repairs allowed.
The NO driving race, get from point A to point B riding on top of NPC cars. It could take a while, so might not be a good option if there are lots of players.