That's not how it works. The only thing that USBKBD.SYS does it to transform a USB keycode (that's just a number for a key on the keyboard) to the corresponding "make" and "break" scancodes as defined by some Microsoft table and then send this sequence of scancodes to the legacy mouse driver (AMOUSE.SYS, MOUSE.SYS) which in turn send that to IBMKBD.SYS (I think, I would need to check the code).
It is the job of upper layers to map these scancodes to a keycode (for example by using the KEYBOARD.DCP table).
And that might also be the problem. It is possible that the legacy mouse driver and/or IBMKBD.SYS do not know what to do with the scancodes or the KEYBOARD.DCP table contains no entry for these scancodes.
Fortunately, for PM virtual keys VK_F13 to VK_F24 are defined which gives me some hope that the OS/2 keyboard subsystem at least knows that these keys exist on some keyboards.