I'd be curious to see if that diagram posted actually fixed the light issue, I don't see how it could.
The key is already in the ignition supplying ground to the key sense circuit before the takeover occurs.
I could be mistaken but it would look as though that diagram is a fix for initial startup (timing) issues.
I'd suggest using the starter output (-) to trigger a latching relay that will stay live and keep the bypass module on the entire run cycle. The issue your having is when the transponder drops out from your bypass somehow the car is car's computer is sensitive to this. Keep it live just that ignition cycle and the light should not illuminate.
Now you have two angles of solution...let us know how it worked out for you.