8BitDo controllers have a few different controller API modes, but they're limited by those APIs. By default they recommend using XInput with PC, but XInput is based on 360 and limited to the set of buttons a first-party 360 controller normally has. This means that the Star/Share button doesn't exist, nor do the additional rear buttons. Instead, they can be mapped to certain functions within the controller firmware. Unfortunately you have to use the 8BitDo Ultimate Software to configure them, and that isn't supported on Linux (doesn't work in Wine either, I tried). There's also an Android version of the Ultimate Software you can try, but I think it only supports some older 8BitDos. There may be some default Star+button combos already, I forget what they do.

If you set it to Switch mode, that enables the Share button to work the way it does on a Switch controller, but software might not recognize it. And the rear buttons still don't exist as distinct buttons since Switch controllers don't have those, they are only ever for macro remapping within the firmware. I don't think there's any way to make them distinct.

Install and run jstest-gtk (It's in the software manager). In the GUI, select your gamepad and you will see a list of buttons/axies that light up when they are used. If it is showing up in there, it means the OS sees it and debugging should be focused on the specific program that you are trying to map keys with. In that situation, you can use something like AntiMicroX to rebind the keys to something your program does recognize.

If the button is not showing up in the jstest-gtk GUI, then the OS isn't seeing the keys and you may need a driver for that gamepad. (If that is the case, one thing to try is to update Mint's Kernel to 6.14, which can be done in Update Manager > View > Linux Kernels)

From what I've heard some of those 8bitdo controller buttons are just remapper on the firmware level and aren't necessarily accessible on the OS level to be arbitrarily mapped.