I assume you already tried to increase touch sensibility in the settings...?
I've got a regular screen protector on (yet), not a privacy one, but it's not working reliably at all too.
There are models with silicone inlet on the spot where the fingerprint reader is, less protective in that area tho, but better for the reader. You may look for a privacy protector that specifically points that out, if any. The thicker/resistant, the less the reader works.