Expecting free birthday dessert from restaurant

Still pretty new to serving, but last night I had a couple celebrated his girlfriend bday and ordered two small items. Once they finished the meal, the guy asked me to give him the check. So I dropped it at the table, then the guy said ‘ where’s the dessert?’ I was confused and said ‘ oh I was going to ask about it but you asked for the bill, sorry. But what would you like to have?’ They told me what they want from the menu then we brought the dessert afterwards. Once they are done, he requested to put part of the cash to the check and the rest with tips on his card. Unfortunately his card only can charge $5 and still have $18 balance for it. I told him about it and then he realized we charged for dessert. He got upset about it and said every restaurant in town offers free birthday dessert. on our menu, desserts are priced individually and if it’s a guest bday, we just gonna put a candle on it. I feel like I did not handle the situation well and left them upset because of it. What doesn’t help is the restaurant won’t waive the dessert for the guest if it’s their birthday which i completely understand.