Well, I've not tested the theory, but here's something to think about.
The new requirement is that to purchase a vehicle you must have a Foreigner ID Number. To get one you must have, or have had in the past, a Residence Permit. You can get a Residence Permit for as little as 1 month which will cost you £40 plus 10 TL for the translation office. (Although it's probably more sensible to get one for 3 or 6 months). So, you get a 1 month Residence Permit, which then entitles you to a Foreigner ID Number and then you buy your vehicle.
I don't currently have a Residence Permit but I DO have a valid Foreigner ID Number.