You typically buy duty free goods at your departure airport, not when you arrive in Belize, or any other foreign airport, unless you're in transit to yet another country. You need to show an international ticket to be able to make duty free purchases (as in you are taking them out of the country). Thus, it is advisable to declare items over allowable limit because if you're selected for inspection in Belize, customs will see what you bought...and assess duties as well as possible fines for lying on your customs declaration! When you depart Belize, you will have duty free opportunities for items you want to bring home.
Have not been in the airport there yet, but this is the way it usually works. When you arrive in an airport in another country, there are usually significant controls to insure you head for immigration, luggage carousel then customs - without side trips to shop.