Here's my reasoning:
HSPU means you are lifting your bodyweight MINUS the weight of your forearms and hands.
Bodyweight overhead press means you are lifting your bodyweight on the bar plus the weight of your arms.
So, a bodyweight overhead press means you are lifting more weight and you will not be ready for it if you can do only a single HSPU. Of course sometimes things sound good in theory but there are things you are missing that you only find out about after you test it.
So, what do you think? If you can do a HSPU does that mean you can do an overhead press?