I just used the service to JFK this morning (writing this from the LH JFK lounge). I got an email from TK suggesting the service, and I booked it online. I could choose from two types of cars, I think it was "black sedan" and SUV; in both cases, it was an SUV.
I believe that the key is not the booking class but whether or not the ticket is changeable. At least, this is the wording on the TK web site. Ex-CAI, K class tickets can typically be changed (either free or for a small fee). Not sure about other departure points.
Shuly