“His being chosen” vs. “His having been chosen”

In either case the verb for should be "...has surprised us." However, here's how I'd write the sentence for clarity and concisesness:

"His selection as headmaster surprised us."

"His selection as headmaster surprised us" could be interpreted as "[The person that he chose] as headmaster surprised us." It's ok if there is no ambiguity in the context, though.

The clearest option is to make it active voice: "We were surprised that he was chosen as a headmaster."

@Except that said: “We were surprised that he was chosen as a headmaster.”
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While this option is clearer, it's actually passive voice since "we" are the receivers of the action "surprised," rather than the doers.

There is nothing at all wrong with saying, “His being chosen as headmaster surprised us.” The gerund phrase, "being chosen," is the subject of the sentence, which is why the pronoun must be "his" instead of "him"; "his" is an adjective describing the subject.

The simple ,the better, I think,

Both sentences seem very passive to me. I think I'd prefer a very apparent, active subject (we, because being surprised is the important word here, right?). Also, as David said, simplest is best.

I don't think Except is wrong in stating that "We were surprised that he was chosen" is more active. "Surprised" is a state of being--more like a noun than a verb--not an action. "Were" is the verb in that suggested sentence, and surprised here is an adjective (modifying we).

Please, correct me if I'm wrong!

When one is writing down thoughts, your brain with its concepts and its power for syntax converts those thoughts as closely as you ordered it in your head. Then it becomes being, and then you think it's having been, because that sounds correct.

The above comments on being "direct" is definitely how you should write.

But, if you must use one of those two mentioned in your initiating question, being chosen refers to describing the headmaster (so think of it as being an adjective) and having been chosen is when you're thinking of the action of when he had actually been chosen in the past.