After a closer look at the circuit board in your photos, the SIP in back of the card edge connector puts 5.6Kohm pullups on all 8 data lines. The 74LS00 chip is powered by the (pulled high) IOBUSINT* signal. The device responds to any I/O port with A7 low.
There doesn't seem to be a 4-bit/5-bit mode switch like the Model I version, unless it's on the joystick itself (which seems likely). Five TRS-80 data bits are being used, so plug it in and see what INP(0) returns when your move the joystick and push the buttons.