What kind of screen protector are you using? Plastic, glass, ceramic?
Does the fingerprint reader unlock the device with just the one finger registered or does it struggle?
You say when you attempt to install another it loops back asking for the PIN, is this after it fails or as soon as you begin the attempt? Does it allow you to start the entry process or does it immediately fail and kick you back to the PIN entry screen?
While it may not be ideal, if everything was working and now is not and the only difference is the screen protector, it would seem that you need a new screen protector. A different brand, material or type. I've seen screen protectors that have a bumpy face, like orange peel, and ones with slightly matted finishes that could cause issues with how well an FPR could see through it, and obviously there would be nothing we could do to improve that. If there was a second protector I would say pull off the one, verify everything is working then attempt to reapply and check the results.