I bought a Taylor suede strap from a friend on another forum for about half price, and love it! I don't know why your Baby, Alice, is flipping when you don't hold it. My Taylors don't do that. Not to be indelicate, but would it do this if you lowered it a bit? Would a slight repositioning of the strap button change its center of gravity sufficiently? (I don't have a Baby, so I honestly don't know, just trying to help.)
Good luck, and let us know what you discover.
cotten