Interesting device. They are sold out at Amazon, or I would get one for testing.
The manual describes the handle as the Directional Pad. Fight sticks do not have analog joysticks, they have what other controllers call the D-pad. The manual also has this statement:
This product does not support the following features.
・Gyroscope, Motion IR Camera, Accelerometer, Player LED, HD Rumble, HOME Button Notification LED, NFC,
L Stick / L Stick Button, R Stick / R Stick Button
We are not going to get an analog joystick out of it.
The D-pad can be encoded in multiple ways. It is possible it is showing up as four discrete buttons, but it could also be encoded as a number 0-15, which the Quadstick is not going to decode properly.
This is the configuration I often use to test random devices plugged into the USB-A port:
USB Host DS4 tester. It brings out lots of the buttons. It would be possible to extend it to decode all the way to usb_2_button_16.
Run that configuration and see if moving the joystick causes some buttons to change while monitoring the quadstick's output on
https://test.quadstick.com or joy.cpl in Windows. If something is changing, note the names of the buttons, B0 through B17, that change when the joystick is moved. We can figure out what to do next.
Another route would be to insert a Brook Wingman XE2 in between the Fight Stick and the USB-A port. Then you would probably see the direction pad show up on usb_1_button_1 through 4, like rows 14-17 in the tester config.