I did this swap but in reverse. Even with the correct airbag the buttons didn't work. Eventually discovered I needed a control module from a flappy padle 'wheel to interface with the airbag.
There was a change in the communications between the two types of 'wheel. Not sure if the same fix would work in placing the 3 spoke in a 4 spoke car.
Richard is the guy who figured it out for me, might be worth contacting him?