There would be a serial number printed on the back of the quadstick.
There is a receipt in the box that contains the order number.
Either of these would do.
Ablegamers does not always order the same configuration. There are over a dozen different sets of "factory" files depending on the options ordered.
Since the PS5 recognizes the quadstick natively, I would focus on troubleshooting the FGC2. Is it in fact a Brook Wingman FGC2? They make several different types of adapters and they all work differently.
Try updating the FGC2 firmware again. Use the most recent NON-test version.
The PS5 requires pressing the "PS" button to wake up the connection. That is typically a soft sip on the center hole.
The Dual Shock USB only settings and disabling the PS5 bluetooth will prevent the DS controller from taking over the connection.
You can plug the DS controller into the FGC2, and the FGC2 into the PC or PS5 to test it independently from the Quadstick.