I never worried about a stuck mouthpiece as long as the instrument would fit in it's case closed. I would let the repair shop take care of the problem with Summer repairs.
On another note I did have a trumpet brought in that was flattened in it's case when the student left it behind her father's pickup and it got run over. I don't know how they did it but the repair shop made it work again and without much cosmetic damage. I told her her Strad was always a bit flat!