sorry i forgot to say my car is 2007 '57 plate 1.6 petrol........still dont see how they can prove a new diff has gone due to looking at tyres!!

hmmm im new to this site as i found problems with 307 gearboxes on google. My car is currently in because the differential in the gearbox peugeot fitted 3 months ago has gone to pot......they are trying to charge me....wait for it.....£2040 to replace.....all because the fornt tyres on my car have melted tread.....they say it is down to me wheelspinning.......well if truth be told i am a tyre fitter by trade with a 3 month old boy i have to pay for so cnnot afford brand new tyres.......i bought a set of second hand tyres off of a car they had been round a track as we all know can cause melting tread......i bought said tyres as the tread is LEGAL which is what matters.........but apprently this is my fault and they will take no futher action until i fork out the £2040 for new box OR £289 to put my car back together so i can tow it away!!!! so in my position what in hell do i do!!!! HELP!!!!!