Hi!
As far as I can see, there is no Hebrew name in the bible starting with FUL (it would be PUL anyway but there is none of that either).
In Yiddish FUL means "full", but "fule levone" (full moon) is the only entry in my Yiddish dictionary that would make at least a little bit of sense. Unless Fulche was just a nickname that had nothing to do with her real given name and she was a little fat as a child. -
CHE would be a deminutive suffix just like in Hanche.
There is an old
German name
FULKE found (if at all) in the very north of Germany. It has to do with "folk" (people).
Now I don't know much about name giving around 1900 (and you didn't tell where your cousin was born and brought up!), but I would guess that if it was a Jewish family, all children would have genuine Jewish names. So I really don't have an answer to your question.
Andy ;—)