I think the covert nature was that Father Pavone just showed up with the medical jet and team and let them know that he wasn't leaving. The hospital probably never figured he would just arrive.
I don’t think they expected this either.
What baffles me—and probably always will—is the question of why they seemed to determined to kill this child in the hospital. What problem did they have with letting this child go home to be with his loving parents when his time came? Who would this hurt, and why did the hospital think it might hurt them?