Yes I would shut one down.
We push to remote hold for slots and keep the engines shut down and then start them without groundcrew supervision so I don't see the big drama myself. If I was going to shutdown 2 and be immobile then I would chat to ATC first just to make sure it didn't cause them any hassle.
A while back we were number 3 in a queue to get airborne when a landing aircraft burst a tyre and closed the runway for 1/2 hour or so. We first shut down one then with ATC oermission shut down the second. Everyone else had to return to stand. ATC gave us 5 mins warning of our departure as we had asked them to and we got away with minimum delay and without the need to fight for a bowser back on stand.