Changing airports is almost never something you planned. Stuttgart Airport to Berlin Brandenburg Airport is 656 km and about 6h 28, and a car covers it in one leg with no change and no ticket barrier between you and your suitcase.
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 correct terminal for your departure — Berlin Brandenburg Airport has 2 and the wrong one costs a shuttle ride you do not have time for. 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 Cologne Bonn Airport, priced the same way.
Luggage stays in the car for the whole 6h 28, 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 Berlin Brandenburg 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.