Beatrice Sank
A Snicket Bear

Those are two options. They are mentioned in the Snicket website or something, though I figured them out with my friend and wrote them in my commonplace book. You also need to think hard. Beatrice Sank makes sense if it is referring to Beatrice the second and telling you that she sank, but also doesnt make sense because the beatrice letters all referrs to what happend to her after SURVIVING her long journey back to civilization. So, if you take BEATRICE SANK and A SNICKET BEAR into consideration, you get BEATRICE SNICKET (what lemony snicket wanted so badly) and A BEAR SANK (which has no meaning as far as I know).

I think that it is "A Snicket BRAE", rather than bear, as that would be an indication as to where Lemony/The Snickets lived or currently live