The Large Binocular Telescope is NOT owned or run by the Vatican.
The Vatican owns the Alice P. Lennon Telescope
https://www.vaticanobservatory.org/telescopes/#alice-p-lennon-telescope
Br. Guy Consolmagno, the Director of the Vatican Astronomy program is an interesting ‘guy’. I heard him speak once at Georgia Southern University. He splits his time between Arizona and Rome.
He has also authored several popular works, including Turn Left at Orion.
https://en.wikipedia.org/wiki/Guy_Consolmagno