I read about that palace. How tacky.

he seems to actually work out of.

It's pretty much all open to the public. The treasures inside are shared with everyone.

near future, after a civil war or coup, it will become a museum too and shared with the public. That is if it isn't bombed to smithereens first. The Pope doesn't need to stay there in it in the meantime. He just needs to meet with the President and be diplomatic. He can scold him about the expense and cost to the people. Popes and other religious leaders have been known to do that in the past. Pompous leaders actually expect it.

meeting rooms, social halls?
I can't imagine the what and why.

could just drive Papa Francis up to that smaller red-roofed building (lower right) and pretend that's all there is to the Presidential quarters.
He could sign the book and make a quick exit.

they could have engaged a decent architect. The thing's a monstrosity.

walk-in closets for a harem?

It's unfortunate but I think if the Pope accepted an invitation to Turkey then he should go.