[MaxD]

If you are going to a push to start button, why not ditch the lock housing.

You need to keep the key and the antenna together for the immobilizer. Wire up an ignition "on" switch and then a button for the starter.

I have not done this yet but this is what I intend to do after the season is over. The Bentley wiring schematic looks simple for this mod.

I can promote it since I haven't done it.

Otherwise the photo reference trygve posted is spot on.