I followed this thread with interest and note what others have said regarding the fitting,the fitter and the cleaning.
The simple conclusion is I'm afraid, it is your fault that the carpet shrunk.
A proper inspection would have for-warned you that it was a BW you can do this with out lifting it a simple burn test would have been sufficient to show it's a polyprop.It also would have revealed the poor fitting which you could have high lighted to the client before starting the cleaning.
As such it would have been apparant that this was a 'walk away' or LM clean only.
I agree the fitters have been no help at all and got in first with her on the accusations front ,which always tips the balance in their favour when it comes to trying to convince her otherwise.
Of course it's nonsense that no Axi's or Wilton can be wet cleaned but this isn't either so the onus is always on us as the professionals to make the right call.
You might get lucky and stretch it back ( but I doubt this) but the gripper will have to be properly fitted now to hold it, which pobably means drill & plugging this is where you can show her how they bodged the fitting.Unlike John I have seen BW double edged fitted on many occasions and you may need now to do this to hold it.
I wish you well but if at the end of the day it all goes wrong, hold your hands up, dig out your insurance and get it replaced for her, it's cheap tat anyway so it won't cost much and you will have redeemed yourself some what in your clients eyes.
Good Luck