I know this won't answer your question...but I know indeed that you can use your factory remote to Auto Start the car. I just can't remember the website & company.
An SC430 had Auto Start & Remote Open on the Convertible top.
He just had to press LOCK, LOCK, LOCK to make it Auto Start.
UNLOCK, UNLOCK, UNLOCK to drop the top.
I don't recall the cost...but I'll be sure to message you if the info comes back to me.