In the really serious world of make-believe, the actors, vehicle, camera, and crew would all be sitting on a flat-bed truck body driven by professionals on a road blocked to all traffic by the police who have been trained, briefed, and rehearsed with the full location personnel. The location has been reserved and used for multiple takes to get the optimal take.
Please Note: Any less preparation than the above may involve significant physical and monetary risk for personnel and property.
That said, this answer deals with the practical lighting problems alone.
There is an order of things to do to control two different lighting situations in the shot.
Start with the windscreen reflections. It's a harder problem to solve than the headlight glare/flare.
To control the reflections on the windscreen, you will need to experiment with a polarizing filter to find the best position. The reflections best limited in this way will be from sources above the vehicle such as the sky and overhead street lighting. Either a normal "linear" or a "circular" polarizing filter will help with this lighting problem. A linear polarizer will have to be aligned for optimal results in each situation.
Removing the windscreen for the shot may be a possibility to consider.
Either way, some attention must be devoted to bringing the interior light level on the car occupants up to the level of the scene to avoid over exposing the car exterior levels.
After you have found the ideal solution for the windscreen, work on the headlights.
Alternative 1. Headlights can be attenuated with a linear polarizer covering the entire lens of each of the headlights in combination with a polarizer on the camera lens (as above). Align the angle of the headlight polarizing filters to get the illumination effect you wish in the viewfinder without changing the camera lens polarizer. This will cut the intensity of the headlights somewhat. The headlight polarizing layer angle is variable. Adjust the headlight polarizing screen until the strived-for effect is achieved.
Alternative 2. Headlights can be attenuated with appropriate neutral density filter covering the lens for the effect you wish.
Alternative 3. Headlights can be dimmed with a rheostat (dimmer) for a DC circuit.
Alternative 4. Remove and substitute different lower intensity lighting for the headlights.
Alternative 5. Do not illuminate the headlights and choose a different time of day for the scene to make the headlight illumination less evident.
For more control of the camera shake, lacking a gyroscopic camera mount, shoot the scene static if you can. Frame each shot so that lack of motion of the background is less evident. Edit the scene so that cut-aways help with the illusion of being shot in motion. Locations will have to be carefully chosen.
Does your school have any green-screen capability to fix the continuity in "post" production?