3 reasons I say pay up and move on:
1) public opinion and consumer law is 95% in favour of the customer.
2) if his bad mouthing you costs you one decent bathroom you could easily lose 10 times what he wants from you. The public love a good moan and with social media he can tell 1000’s of local people. It doesn’t matter you’ve done nothing wrong
3) best case you end up in court, you’re business ended up in the press, again the plumbers always guilty. You loose nights of sleep, days of messing about with court dates letters etc and incur costs from solicitors etc.
Anyone would want to stick it to him but the best decision for your mental health and business is to pay what he asks. You could try offering him half?
Difficult to give advice when we only have part of the story.
The customer could be a rogue, or he could be a reasonably decent guy who genuinely had a leak.
Unfortunately the OP didn’t go back to verify what the truth was.
The law is also fair for a plumber as long as evidence is in the plumbers favour. The customer would have to prove it was the plumbers faulty workmanship and responsibility.
Small Claims Court should be okay for trades people if they have followed all proper procedures with dealing with customer, including overall good quality work and having all evidence.
Problem for the OP here is he wasn’t able to deal with the alleged issue.
If the OP does pay the money, the customer could still bad mouth him and it would look to everyone that the plumber had been liable.
As I say, it is difficult to know what to do. If customer did have a genuine problem caused by the OP and asked a sum of money to a reasonable amount and without showing aggressive intent, then am sure sometimes best to keep them happy, pay and walk away.
I remember being involved with a friends customer who was a con woman. She tried to claim for damage to a vinyl floor actually caused by herself. That sort of person is best to never be handed any money, even just on a principle.
She thinks of insurance, including trades people’s, as something to furnish her lifestyle.