I am afraid to take the back off.
Unless there's a Hamilton marking on the outside of the back of the case, that might be the only way to determine whether it even is a genuine Hamilton watch. It being a slow afternoon at the office, I looked through a bunch of catalogs, mid 1930s to mid 1960s, which was sort of the golden era of these tiny diamond set cocktail watches. I didn't find a match. What I suspect you may have is a Hamilton movement in a privately made aftermarket case. This was a common practice. Jewelers would buy either a lower-end gold filled model or used Hamilton watch, scrap the original case, create a new precious metal and often diamond studded case and resell it.