You certainly don’t have to write out the whole name of the event every time it comes up in your story. In fact, you might not even have to name the event at all. Anthony Doerr’s novella “Village 113” takes place in a village that is soon to be submerged as a result of the building of a massive dam. Indeed, China’s Three Gorges Dam did just that, but Doerr doesn’t have to name it to capture the essence of the experience. Instead, Doerr relies on strong details throughout the story to suggest this. If you put readers firmly in the era of 1939 and surround them with the anticipation and sensory detail of the fair, you may not need to name it at all.
If the story must refer to it by name often, find out how people talked about it at the time. Indeed, the full name was likely on posters and marketing material, but perhaps there was a truncated name used in casual conversation. Stephen O’Connor’s short story “Another Nice Mess,” uses “Great War,” the term contemporaries used for the First World War. Also, look into other language used in association with the event. O’Connor does this, as well, indicating the soldiers wore “doughboy uniforms.” This will help situate the reader in place and time and lend your work an air of authenticity.