Modelling Spring Powered Drive Train

At first I thought you meant real modelling (as in making stuff)... then I realised you probably mean mathematical modelling.
I'd say you need less mathematical modelling and more building and experimentation.
All the modelling in the world won't help if you just get wheelspin.
Just turn up a stepped rear axle to wind your string on an try the different diameters.
An ounce of experiment is worth a pound of maths IMO. _{(Other units of mass are also available, terms and conditions apply)}
Del

Along with the mass of the whole dragster, there will also be the moment of inertia for all of the parts that rotate. Don't forget your rolling friction and air drag in your equations. I would certainly perform a Laplace transform on all of your initial equations so that you can easily combine all functions before doing a reverse Laplace transform. This will be much easier than doing all of the convolutions you'll have to do.

It is very unlikely you are going to find a "linear" spring, with constant k, unless you are working with fairly small changes in extension. Sounds like you aren't. Something like a clock spring might give you more constant k.

Also, a picture is worth a thousand words (unless, of course, the words are on the Internet, in which case the picture is worth at least 100,000 words).

If you have a 3D CAD model and are looking to simulate the dynamic action of the whole thing, with an ability to change spring parameters, inertia and such "what if" scenarios, please take alook at ADAMS or visulaNastran 4D software. The latter is defunct, so MSC may give it to you free if you ask them nicely. There are some videos on YouTube which demonstate the software.

Use a spread sheet: ideal for this sort of problem.