I am not sure when this changed, but a few months ago I was able to change the Stadia controllers bindings by using the PC UI. I suspect it happened in the latest update.
The method for changing it was as follows:
1. Start game in Chrome, login ad choose character to change bindings on
2. Using keyboard and mouse, open the Controls menu
3. Choose the binding that should be changed
4. In the dialog, press wanted button on the controller. The dialog would then display the button pressed
5. Click "Bind" and the change would be saved
Now, when pressing the controller button, the "Controls" UI just closes and performs the action mapped to the button, for example, a light attack if R2 was pressed.
I found this functionality/workaround essential for playing the game, since the default layouts are not optimal. So, I really hope you can reverse/re-implement this functionality.
Thanks!