That's a new one on me. I'm pretty sure there is a starter relay in the fuse area under the hood. That might be worth looking at. In my wife's GT, there is another relay just like it you can swap around.

I wouldn't rule out the ignition switch. We've had three GTPs and one GT in the family and have replaced the ignition switch 6 times in them for various weird issues from trip computer issues to actual delayed starting. (holding the key on for a second before anything happens).

Good luck.