As has been said, I would be very wary of cleaning perspex, it's much softer than glass and presumably scratches much more easily. If you ever want to clean any in future, get the customer to sign a disclaimer.
Did they tell you it was perspex before you started?
Firstly I would report it to my insurance company with all the facts including that they didn't mention it till three weeks after the clean, then wait to see what the insurance company says. If they accept the claim then go ahead on that basis. If they decline responsibility you report that back to the customer. That should be the end of it, it would be up to the customer to argue with your insurance company - that's why you have insurance.
If you still have problems, the below might be useful:
I am quite sure that 99% of cases of WFP 'scratching' glass is actually pre-existing scratches being exposed by the brush removing soiling that had disguised the scratch, and the customer giving the glass much closer scrutiny to see if the system has really cleaned the glass.
I'm so confident of this that on the very few occasions any of us have been accused of causing scratches, I have invited the customer to try to scratch a window with one of our brushes on the promise that if they succeed we will replace the glass we have been accused of scratching, together with the glass the customer has succeeded in scratching.
To date we have only ever had one customer accept the challenge, and that one failed to make the slightest mark on the glass.
They still cancel 
but at least if they try to tell anyone else it was because we scratched their glass we can demonstrate they were unable (or unwilling) to prove we were responsible.