Don’t cancel the order pal. The payment will still come to you reputation or not. Unless of course it’s disputed, but in my experience this is a very very very rare occurrence if your customer contact is good and you keep customers updated should you experience any issues.
Customers often can’t get back into their anon order once they have left the screen, and you also have to respect someone’s right to privacy in my opinion. If the facility on the website is here, you have to honour it.
Best of luck