Do NOT track the car in the current condition. You will be pushing the car to 9/10ths, and if there is a minor problem then this can very quickly become a big problem if something lets go at high speed. You risk not only your life, but the lives of your fellow trackday goers.
Now the 'non-positive' rant is out of the way, some more practical advice!
If budget is your concern, and it looks like it is with only £4K to go play with, then I'd consider getting an early MX-5. Strip the car, do the brakes, add roll over protection, and most importantly if you have a bad back (previous fellow sufferer here) put a supportive bucket seat with harnesses in. That will give you a very fun little car that is no slower than a Zed for less than £2K, and if you write it off at the first day then you've only lost that much. It would be very easy to do that much damage to either a Westie or a Zed if you had a minor off, so worth considering another car from that POV.
Westies are huge fun but if it's a very cold day, or if it's raining, you're really not going to have a very fun day. Something like a '5 or MR2 or even the Zed if you do decide to go that route will be far more enjoyable (and quicker) if conditions are bad. The Westie/'5/'2 will be cheap on consumables, the Zed much less so.
None of the cars that have been mentioned would be a bad choice for track use, just depends on what you want out of it.