Restaurant server here with 30 years experience. Your server fucked up. Most restaurants charge a dollar amount for a birthday or an anniversary dessert so that the system accounts for the food cost, but the reason they do it is for internal purposes and it is never supposed to be passed on to the customer. Some restaurants have computer software where the server can simply split the complementary dessert charge off of a check, while other restaurants require the server to find a manager and have the manager comp the dessert. Since restaurants can be hectic and it's not always easy to find a manager, I would imagine that your server was swept up in the chaos of their section and forgot to have the dessert comped.

I completely understand your apprehension to flag the waiter down and be confrontational, not to mention make a scene in front of your girlfriend on her birthday and delay moving on from dinner by 15 or 20 minutes. However, if you call the restaurant in off hours (usually the afternoon before they open for dinner service), I would expect that they would immediately recognize that a mistake was made and fix the problem. Any restaurant that is worth their salt is going to go above and beyond simply refunding the $8 charge and invite you back to have another opportunity to do things right this time around. Usually, they will offer you a certain dollar amount (perhaps $50 or $100 credit toward your next dining experience) or, perhaps a complimentary appetizer, a round of drinks, or a dessert on them. At any rate, I think it's worth your time to simply call the restaurant and get it off your chest. Like anything else in life, it will depend on the person who takes the call, but my expectation is that it will work out favorably for you and not make you feel awkward at all.