When did Fraser become Frasure?

I'd suspect that it might not even be Fraser, given that there are only a couple thousand people with this name. It's an oddity, given there are three more common ways to spell the name.

You'll find that most genealogy sites worthless for such a small name, as all you'll find is the usual machine cut and paste answer.

Judging from the earliest records for the name that I see, it looks like a Kentucky illiterate version. So I'd conjecture that they were likely part of the large number of Frasers and Munros who willing or unwillingly immigrated after William of Orange's ascension. You can kind of see a parallel between the transformation of Munro to Monroe and the change from Fraser or Frazier to Frasure.

At a guess, ignorance, illiteracy or a desire to be separate from everyone else. (I have an ancestor who changed from "Adams" to "Addams" to be separate from everyone else.)

You'd have to trace your Frasure/Fraser ancestry to see who made the change. The chance of someone knowing your family and seeing your question here in the 18 hours it will be on the first page is slim.