Before you pay the additional $600, you should ask for itemized statements for the vet bill(s) and for the "confidence building" class. Moreover, you should ask for proof of completion of the class.
Upon being presented with the itemized statements, you may wish to write a letter to him indicating that the enclosed check represents payment in full for any and all expenses incurred as a result of the incident on xxx date. You should pay a certified check and in the memo section put "paid in full for incident on xxx date."
If you keep paying bills without getting verification, the owner of the dog that was bitten will keep finding ways to milk money from you.
You may be responsible for the vet bills, and MAYBE for the class, but beyond that, you should NOT BE RESPONSIBLE for any other "damages" as a result of the dog bite.
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Unless he produces verification of the vet bills and certification of the completion of the "confidence building class," you shouldn't pay him. He could sue you in small claims
court. However, if he does, he will have to produce verification of the vet bills, etc.