No! "Master stroke" is a different thing; and in any case, he does not say that.
He says: "Your housemaster stroke monk is doing the religious bit .."
Here, "stroke" just means "or"... You'll also hear "slash" being used like that.
Thanks for that, both. I went straight to 0:37 on the video and only heard the "doing the religious bit". I didn't hear the words in the other part of the question anywhere so (wrongly) assumed the OP had heard them correctly.