That's terrible
A good example of why I want nothing to do with BT and was glad to get my FTTC and telephone from Sky.
I don't think you should have paid a disputed invoice. By doing so, it may be taken as an admission of your liability.
In any case I think your next stop is legal advice if you can get any without losing an arm or a leg. Otherwise it's CAB.
Of course we mustn't forget that if BT have now had your cash then where is your FTTC. Get them to install it. Failing that claim a refund from the CEO, whoever it is these days.