Using the Controllable mod with Create-Tweaked-Controllers enables the player to move while using the tweaked link controller.

For this one I think I will have to see with the Controllable dev on how to implement that, and whether or not this has to be done on my side

Here is the issue I made in the Controllable mod's repository : MrCrayfish/Controllable#445

Update : I managed to integrate Controllable on my side and cancel the required events.
The drawbacks are that you cannot move or turn while using the controller, and also I made that if you press your use button (usually left trigger) it will exit the controller. This makes it that one input is unavailable unless you bind it to something else.
The "exit controller" keybind is shown while using the controller

This change will appear for the next update

Awesome, thank you for the quick response.