where to put the possessive "s" with an appositive parenthetical [duplicate]
There's nothing wrong with the possessive marker at the end. Compare The king of England's crown, where the -s is at the end of the noun group.
It does sound a bit more awkward if you have a whole apposition following the head noun, but syntactically it looks fine to me.
Or you could write It is my gym instructor Anne's birthday.