Now it may go against the grain here, but I’d approach this differently.
I’d firstly find out how much the repair/replacement cost is (not from the customer, look it up yourself)
Then, you have to decide how much you want to keep the customer. It’s a simple decision.
For example, if the repair cost is £20 but the customer is a great house that’s well priced and easy to do, and worth keeping even after a £20 loss, then it’s simple. I would just tell the customer I was not sure it could have been my fault, but regardless I’ll pay to replace anyway.
The customer will most likely be over the moon with that attitude, especially if they were expecting a dispute and an argument, and you’ll most likely keep them forever.
However, if you’re not bothered about keeping the customer or the repair cost is too high, you could challenge them to prove it was your fault. You’ll almost certainly lose the customer, and might end up in a pain-in-the-neck dispute.