I have not been able to find a copy of The Menace (1932). There is a photo of one hat on IMDb, which I have shared here. I do not know if there are any other hats.
Of Bette's performance, VARIETY said, "Bette Davis has to take a decided second to Natalie Moorhead as the adventuress who kills her husband and blames it on her stepson." In her autobiography, Bette merely said, "I appeared in a monstrosity called The Menace opposite Walter Byron and H. B. Warner. My part consisted of a great many falls out of closets. The picture was made in eight days. I knew I had obviously reached bottom" (Davis, 118).
The hat was also featured in the February 1932 PHOTOPLAY. The film was tentatively titled The Feathered Serpent.
Davis, Bette. The Lonely Life: An Autobiography. G. P. Putnam's Sons, 1962.Ringgold, Gene. Bette Davis: Her Films and Career. Citadel Press. 1966, 1985.