In short, walk away and consider it a lesson learned.
In long...
First off, although this will now sound like a huge criticism of you it’s really not meant to be. Nothing you’ve done sounds particularly wrong, and thousands of people do the same thing all the time and everything is dandy. Hell I’ve done the same as you in the past, so I’m not judging your actions at all. However, you’ve now sadly discovered the main issues with having a friendly garage and also with supplying parts yourself.
From a clinical POV, the garage is right. You supplied the parts (granted they were fine), you supplied the instructions, you negotiated a rate direct with their employee, you lost out on transport costs for the first trip etc. Yes they bodged the install a bit by the sounds of it, but ultimately who was your contract with? Sounds to me like it was with the employee, so your claim is against him personally, but then it depends on how you paid (cash?) and if you got a receipt. Again, it’s just bad luck here.
Every one one is a mate until things go wrong, and then suddenly they look out for themselves. The garage has done you favours in the past to keep you sweet as a customer, but now things have gone wrong and they are at risk of losing money they’re going back to black and white terms. Do I think it’s lousy practice? Sure. Do I see where they’re coming from? Yes.
Ultimately if you took this to court then the waters are muddy enough that I wouldn’t be comfortable saying you’d have a good chance, or any chance. It’s time to be more practical, and going forward be very careful about supplying parts, make sure you have everything in writing about the work being done and the costs, and accept that whilst jobs may now cost you more at least you’ll have protection when things go wrong. Like I said, we’ve all been there and I feel for you, but whilst there’s no harm in asking for money back from your garage I strongly suspect that ship has sailed and they’re done with you, as you should be with them.