There's no need to use a phonetic spelling of
Persephone. Anyone who doesn't get it the first time can easily get it when explained. It's not that different than
Penelope or
Hermione.
Personally I strongly dislike phonetic spellings in general. They always seem as if the parents either are illiterate or think that I am. I know you are not. But your dd won't have the benefit of this post / thread to explain how her parents came up with such an unorthodox spelling. Let people learn something. It won't hurt them.
There are a few phonetic variants that don't bother me, especially for Irish / Gaelic names. But even then I only like those phonetic variants with a strong history behind them. Afterall, they came into existence before the internet and the age of information. It makes sense to me that an English person in the 17th century might turn
Maire into
Maura or
Moira, or
Diarmuid into
Dermot, etc. But I have much less patience with modern, random, individualized takes on spellings. I'm persnickety that way.
I realize you are trying to make things easier for your child and the rest of us, and I give you credit for that. I just don't think
Persephone is so "out there" that people can't catch on. :-)