You should be able to get vehicle, height (to a max height), perhaps number of wheels, location/shape of windows (if the beams go through the windows) and the general shape.
You can probably just have a template (or a few templates) for what the side profile of a car, truck, van ect look like. You can then stretch each templates to the dimensions measure and subtract the recorded shape from the template shape. The template with the least difference is the closest match. This can be improved by allowing the shape to be more variable. For example, the height of the hood could be moved up or down to some degree based on min/max recorded ratios of hood height to roof height. If you have a collection of such ratios (or actual recorded values if you find them online) and templates, then you should be able to do well enough. You could get these ratios simply by analyzing a number of vehicle photos.
This should work fairly well overall if you have good, representative templates and aren't trying to be too specific as to what the vehicle is. For example, finding templates that you can use to tell the difference between an crossover and a van might be difficult, given how your system is stated to work, but should work fine if you allow for a bit of leeway as to what a crossover is classified as.
Edit:
Actually, you could use a single template and just have a few adjustable points (up to around 10 such points), the configuration of which could be used to classify the vehicle. A few examples:
- Start of the hood
- Hood/windshield intersection
- Roof/windshield intersection
- Tire/body intersection (2 such points for each tire)
The result would be a blocky but fairly accurate vehicle shape. Roughly where those points are and if they exist at all should be useful for telling vehicle type. Although, having fixed templates would be much simpler and if say a van is listed as a truck, you could probably use that van as an additional template for a van.