If you read the OPs posts carefully you will see that the vet looked after a number of horses belonging to different clients (who are not all present at the time) in that visit. You will also see that the OP accidentally tossed Invoice #1 (March dental) and in the middle of June their trainer informed them the vet was complaining about an unpaid balance. The OP had no other invoice before then - no outstanding balance invoice for April, or May. No phone call until after the OP called after hearing from the trainer. No email until after the OP called and asked for the information on amount owing.
Your vet wants to write up the billing at the office later (and probably have an employee handle the actual pricing/invoicing/records update/etc in order to spend less of their valuable time on minutiae). Why do you think this is unusual? Especially for a vet dealing with a trainer’s client owned group of horses.
I don’t know about your vet, but vets around here don’t have set rates for dental care. It varies depending on how much work is needed, how much sedation is used, etc.
I’m not sure how the OP doing stuff wrong means the vet clinic didn’t do anything wrong. The OP did something wrong in tossing the invoice, but the clinic was also wrong in how they handled it. If that original invoice had been lost in the mail, no one would be blaming the OP because the clinic failed to contact the OP prior to telling the trainer. If the clinic had sent an April outstanding balance invoice the OP would have paid it weeks ago. Basic Business 101.