As a guess -- when the vehicle is on the track, I'd do everything as 2D. The track looks 3D, but that doesn't mean the vehicle physics have to be.

Basically the car is always on the surface of a rectangle. For the "tube" sections, the left/right sides of the rectangle are teleporters to each other.

When the car flies off the end of a bit of track, then vehicle physics are disabled and regular 3D projectile physics take over, until it hits another bit of track.

Agreed with Hodgman, in this case reducing the dimensionality of the problem is a good bet. The same representation also simplifies things for AI drivers. Shaun Hargreaves posted some stuff on his blog about a motorbike racing game that used integer coordinates for the track and relied on overflow to correctly handle the circular nature of laps. Maybe you could leverage the same thing in both directions for the wraparound physics on the tube sections? (I'm sure you'll find his blog with a bit of Googling)

On a side-note though, I always thought the best physics model in a similar game was in Wipeout 3 for PS1, (Wip3Out if you prefer). That did have 3D physics in the sense of the cars floating above the track, and also included nose-up/down to allow the cars to 'fly' a little bit and air brakes used for drifting or tight corners. Recommended if you haven't played it. (The other titles in the Wipeout franchise are all similar but the physics model is subtly different in some of them. Wip3Out nailed it, in my opinion, it's the 'reference' Wipeout model for me.)