I was a twenty-something in the seventies and like Jo and Gyppo, never heard the word. Of course, I am in the U.S., not England so there are numerous words unique to both sides of the pond. Closest word I can think of, in this context, is "bestie" meaning "my best friend" but that word is more modern, a part of the texting world, e.g. BFF. The word bezzie sounds Aussie to me. Are any of your characters from down under?
AQ
Edit: how about "Sunday best"?