Remember how messy it was for DePaul re: Ray Meyer when they fired Joey? This should be worse, so I would basically go completely outside the "family" if I'm Georgetown.
I really don't know what Gtown is willing to pay, so it's pretty hard to speculate. But with the Big East being good now without the Hoyas, the next coach better get the job done or they risk becoming irrelevant.
Today's kids know nothing of the GT-SJU-Cuse conference; their field of experience is the Nova-Butler-Creighton conference. Georgetown was good in their parents' day. Fix it fast or it becomes ingrained.