There's a very simple test for whether a bishop is an employee of the Vatican or not.
Does the Vatican pay him?
The answer, of course, is no.
We would not be having this conversation if the church were organized all under one corporation. That is why I posted Restatement Section 220, which applies where one corporation, as the "parent" or "master," may be required to answer for the tortious conduct of the "agent" or "servant." This is called respondeat superior liability, which imputes liability from one person or corporation to another.
There's a very simple test for whether a bishop is an employee of the Vatican or not. Does the Vatican pay him?
Who pays who and how is only one factor to be considered under the Restatement Sect. 220 test. No one factor is outcome determinative. So it is not as simple as you say.