Is there a name for this accent / speech style used by the female char in this clip?:
I believe Mae West had a similar way of speaking. Mid-Atlantic by any chance? 
I’m a mid-Atlantic (Maryland) native, and I don’t hear any distinctive accent, certainly not one characteristic of, say, Baltimore, or any other area known for its accent. There may be a faint Southern twang, but it’s very weak. (She says “'em” for “them” a couple of times.) The actress, Carole Landis, was born in California.
Mae West, who had a very distinctive style of speaking, was born in Brooklyn, but I think her “sound” was more a mannerism than something characteristic of Brooklyn or New York.
The only thing I notice in her accent is that it’s typical of certain movies of that era. Perhaps it’s not a real accent at all but one only used by actors of that time.
Not only don’t I hear an accent, but I don’t hear anything similar to Mae West.