You could use AutoHotKey which has macros to bind Joystick button presses and you can then simply trigger a keyboard keypress or execute a command or something like that:
A simple macro like
Joy3::Send #!{PrintScreen}
should do the trick (Win+Alt+PrtSc, default keybind for screenshots with the Gamebar). You just need to find out which button AutoHotKey has to react on.
Here’s a list of the Joystick identifiers that you can bind, there is also a testscript to identify a button when pressed: