I've been thinking about the barycentre again, and I now think that Sawicki and the BA are wrong when they claim that a person at the Earth's CoM doesn't feel the force of the Moon's gravity
because they are in free fall about the barycentre. They
are in free fall about the barycentre, but that's not why they don't feel the Moon's gravity. The correct reason is that they are also in free fall about the Moon's CoM!
I haven't done (and perhaps can't do) the maths, but
Starry Night Pro clearly demonstrates that if you take the Moon's CoM as your fixed frame of reference, then the Earth's CoM does indeed revolve about it in a neat elliptic path, and is therefore in free fall (like all objects travelling through a gravitational field along a path that is a conic section).
If this analysis is correct, then
my objection to JohnD's first post is no longer valid. The fact that the barycentre is between the tidal bulges is irrelevant to lunar tides. What is relevant is that they are both on the same side of the Moon's CoM.