Don't pay them.. just because you were last to fly it doesn't mean that it is your fault. What happens when a plane goes in for a 100 hr inspection? With their reasoning no one would want to fly the 99th hour as it would be a pretty expensive exercise.
Don't admit fault or liability but them to put down in writing their demands and get them to answer the following questions.
What is the part number of the spinner and backing plate.
Is it an OEM or PMA part.
Also ask them how often they dynamically balance the propeller and supply recorded references to the work carried out.
What torque was applied to the screw on installation and can you see the calibration records for the torque wrench.
Who was the person that put it on last and what date and time it occurred.
What is the part number of the screw and what approval are they using to use stainless steel hardware.
This will get them off your back. If not tell the local FAA inspectors as well...