I never would have thought /ba-TAY-uh/, that's for sure (How the heck did she get /TAY-uh/ from the latter part of the name? I don't see how -te could be /TAY-uh/. Hmm.). My first thoughts were /BEE-tuh/, /BEE-tee/ (like Beetee in HG), or /BEET/.
I have never seen or heard of Beate before. On first glance I thought it looked like a possible nn for Beatrice.
According to Behind the Name, it is pronounced /bee-AH-tuh/. German form of Beata ("Derived from Latin beatus meaning "blessed". This was the name of a few minor saints.").
I taught in Tanzania for a month during college and a girl from my host family was Beatrice nn Beate (Be uh tay).
Re: Beate