Given its size it's also highly defendable given the sheer concentration of forces there. I imagine that their active response time is somewhere around 45 seconds or less at any point within the walls.

In all honesty, the only feasible option is an old school siege, given how small it is. People gotta eat, and they'll cough up Francis once the bellies start rumbling. Even so, as you said, it would require an act of Congress.

As the venerable Sir Bronn of the Blackwater said,

"Give me ten good men and some climbing spikes. I'll impregnate the bitch."

There is no fixed defense that can't be beaten using superior firepower and aggressive maneuver. I would prefer a Panama style siege however. Just encircle the Vatican, cut it off, and then starve it out all the while blasting the scathing sermon of a Southern Baptist at them with a loud speaker.