Where to source replacement joystick

The bad news is Marshall do not sell it as a standalone spare, which is exactly the problem you have already hit. The joystick is a small tactile SMD component soldered directly to the control PCB inside the earcup.

Your best options in rough order of ease:

The donor headphone route you already know about is honestly still the most reliable path. Keep an eye on eBay for Monitor III ANC units listed as faulty, no sound, broken headband etc. Anything except joystick fault. You can usually pick them up for not much and the joystick will be fine.

The other option is sourcing a generic replacement joystick directly. The component itself is a 5-way tactile switch, also called a navigation switch or hat switch. If you can get the board out and read any markings off the existing joystick, you can cross reference it on AliExpress or with a component supplier like Mouser or Farnell. The catch is you need to be comfortable with soldering at PCB level to swap it.

Marshall's own support is worth a contact too. They are better than most audio brands for spares and occasionally sell service parts that are not listed publicly. Worth an email directly? I've had them send over a replacement piece before and they only charged shipping