I can see why the OP is miffed but I still think they should seriously consider whether it is worthwhile.

I am mystfied as to why the dealer or the OP ever spent any money on this car in the first place. It wasn't financially vaible to do the gearbox IMO and it was clear it was a duffer. The dealer should have just taken it back, maybe sold the OP another car and punted it through the auctions. That would have been better, and cheaper, for both parties.

The situation is a bit of a mess as the OP has done work not authorised by the dealer so you have third parties in there who may or may not have diagosed what was wrong correctly.

If it goes to court then the OP will probably have to get engineers reports as to what exactly is wrong with the car which will take time and money. The dealer may dispute them which will mean more reports. This will all take lots of time and upfront expentiture which the OP may, or may not, get back. The court will try and force both parties to come to a deal and will push you to go through arbitration first which will all take more time.

Months later, it could end up before a judge who will probably try and find a compromise and force both sides to accept a deal which neither are particularly happy with.

The whole thing has clearly been a disaster for both buyer and seller. The seller has probably lost money, the buyer has lost money. Unfortunately this is the world of cheap old cars - £1300 doesn't buy much car these days and they are a major risk for both seller and buyer. That is why they are cheap - because they are a gamble.

Technically, the OP is justified in feeling badly treated but whether it is worthwhile persuing this for months and spending yet more money when they may not get everything they want from a court case anyway for the amount of money involved is questionable.

Sometimes it is best just to put something down to experience and move on. The experience is don't buy a cheap Mazda 6 diesel - a quick read of this forum before purchase would have told them that!