warranty doesn't really come into it. Lexus Derby sold you a car that was defective, so who they get to pay the cost is their problem, but they are responsible to fix the car.
Incidentally, did you take any HP for the car? If so under the consurmer credit act the HP company is also equally liable to you, so you can attack on both fronts. I successfully did this with Ford many years ago when they refused to do any permanent repairs on my car, but simply fix under warranty until the warranty ran out, then i would be on my own. I took the case up with Ford Credit and after 3 years and no further payments they agreed to accept the car back. point being that you can chase on both counts.