@Drew what does that paper have to do with it? I don't see any mention of the monadicity theorem in it.
In the book "Toposes, Triples, and Theories" by Barr and Wells, it is very definitely labelled "Beck's Theorem". So Barr does not seem to think it is the "Barr-Beck theorem".
@JonathanBeardsley I don't know references. As Tom suggests, I would take some formulation which works in simplicial sets and see if you can extend it to simplicial sheaves.
