You should try and get written cleaning instructions from the manufacturer/supplier. If it states HWE then you might have grounds to fight this.
I doubt it,
It may well be wet cleanable but they will argue and win that it was not cleaned professionally,
It their view it would be up to the carpet cleaner to have the necessary expertise to be able to know how much moisture can be applied.
Your carpet fitters were right, this carpet can not, and should not, be power stretched back, there is not enough body in the carpet and it will almost certainly open up or at least pull off the griper again leaving loose yarn. as already shown, if the carpet is attempted to pull back on to this it will just pull the pile away from the body of the carpet. You don't want this as exposed gripper is a hazard, they will blame you if some has their feet ripped open, it was probably another reason why the fitters turned it down.
We have cleaned loads of this carpet and its rubbish, its a low moisture clean only using pads and as such it just glides across the top surface, the end product is not great and we always qualify this with the client.
Its not £65 -£70 a sq mt job whether its from CT or not, £15-20 at best.
Get your insurance involved, put it down to experience and move on, if not you will be for ever going back and forth to this customer and neither of you wants that