$3000 for a $270 module and less than two hours labor.

You took it to a dealer, what did you expect to happen? Were you expecting honesty? Good customer service?
Or you were expecting them to price it such that you'd trade the car in for an overpriced new car they happened to have on the lot?
Because that is what happened.

As to the car "shuddering" and then the light was on - well, sounds to me that your battery is on it's way out. This same thing happened to me and when a battery in these cars goes, it tends to take modules with them. My airbag light is on for the same reason, someday I'll tend to it.

I keep preaching "replace your battery at four years" but not only will no one listen to me, this time, I didn't even listen to me.
German electronics - the worst thing, perhaps the ONLY bad thing, about our beloved cars.