If cleared to cross a runway and expedite but are slow to get away, are you obliged to stop?
If you feel you cannot safely comply with any ATC instruction, you can simply reply “Unable” and the controllers will amend your clearance.  Remember: you are pilot in command; you are the final authority on the operation of that aircraft.

So the scenario might play out as:

ATC: “Bigjet, cleared to cross Runway 36.  Expedite crossing and report clear.”

Bigjet: “Cross runway 36 and report clear, Bigjet.”

Bigjet: “So-And-So Ground, Bigjet, Unable.  I’ve got a problem here.”

ATC:  “Bigjet roger hold short of Runway 36 for arriving traffic.  Will you be able to taxi after traffic clears the active?”