The 268 km between Stuttgart Airport and Frankfurt-Hahn Airport take roughly 2h 48 by road. Booked as a transfer it is one vehicle from one terminal kerb to the other, at a price agreed in advance — which matters more than usual on a day that has already gone wrong.
The pickup is driven by your inbound flight number: the driver watches the actual landing time and waits 60 minutes free after touchdown, which is what a diverted or delayed arrival needs. At the far end the drop-off is at the terminal door, not the car park. Tell us the departing flight at booking and the schedule is worked backwards from its check-in deadline rather than forwards from a convenient departure. We run the same journey to Düsseldorf Airport, priced the same way.
Luggage stays in the car for the whole 2h 48, which is the single biggest practical difference from any rail route: nothing is lifted onto a train at Stuttgart Airport, off it again at the interchange, or onto a second one for the run into Frankfurt-Hahn Airport. If the airline has rebooked you and is reimbursing ground transport, the invoice shows the route, the date and the fare in a form an airline will accept.