Update on this. I never charged the guest because, luckily, the builder did not charge me to fix the door. He was doing a complete renovation on two bathrooms, costing several thousands of pounds, so never asked for extra to sort out the door.
Since then, I have had another guest break my very expensive front door lock (over £400 to replace including labour). Accidents do happen, but this time I did expect the guest to pay as the damage was caused by him ingoring my instructions and being very brutal with the door! He absolutely did not want to pay anything, but luckily Airbnb sided with me in the resolution centre. I have no idea if they took any money from his deposit or paid with the Host Guarantee insurance, as that information was not provided.
I'm still happy to let small damages pass. The other day a guest left black marks on new Egyptian linen sheets, which won't wash out, but she told me straight away and offered to pay. Most guests just keep quite about these things and hope you won't notice. Plus, this guest had already bought me flowers and a bag of (expensive) coffee to replace the small amount she used. I'm willing to put the sheets down to wear and tear for a considerate and honest guest.