Sharing gen 3 magic track pad with multiple MacOS devices

I saw this thread of a user being able to use Toothfairy to better share/swap a magic track pad between devices. It apparently does not work for gen 2. Would this work for gen 3? I haven’t bought the app because this is the only thing I would need it for, so I wanted to check first. MTPs being able to only connect to one device at a time is super frustrating for me, requiring me to forget and re-connect each device every time so I’m hoping to find an app that can at least somewhat automate this to more of a one-press solution.

My understanding is that there isn’t really a Magic Trackpad generation 3. The current model is the same as the Magic Trackpad 2 but with more rounded corners and different color options. My presumption is that it does not support pairing with multiple devices.