My apologies for not replying to you sooner, the truth is that i wanted to play on Linux before replying so I could give you feedback for both OS. ;-)
And now I did, I just played on Linux, and I have to say...
The game in unplayable with a controller on Linux, I tried to save the mother and I had to watch her falling into the pit because the game didn't respond to my controller!
Here is my detailed feedback, the game recognizes the controller, I can move the girl in all directions and save the father and the sister.
But I could do it only because I previously played with my controller on Windows and I remembered that I had to press X in order to cut out the wines and shoo the dogs.
Because on Linux, the game wouldn't tell me which button to press, it says "Press space" but I am not playing with my keyboard, I am playing with my controller!
So I think you might need to make sure that the game indicates the right buttons for the right device.
But that's not all, the game will not let me save the mother with my controller!
I remembered from Windows that I had to press the right trigger (or was it the left?) in order to pull her so I did press the right trigger in order to pull her.
And...the game didn't react, it didn't realize that I was pressing the button.
So I tried every each button on my controller while the mother was begging her daughter to help her and the game didn't react to any of them until she felt into the pit without I could do anything about it.
So you might also need to make sure that the game reacts to controller in the mother rescue on Linux.
And about the menu input issue, it is not fixed nor in Windows nor in Linux, with both OS, the game doesn't react to controller's inputs in menus!
Since you seem to not know how to fix this issue, may I suggest you to talk about it in Godot community so other developers might help you to sort it out? :-)