So... I had a variety of people (including Cell! ) tell me about the Shape->Move Shape option in body slide. [i think this should be mentioned in the tutorial? But it's not quite enough...]

So, first I need to open the nif file in nifskope, and locate the Translation values on the NiTriShape nodes. They are all zeros for the "body" but they are something else for the clothing (I added a red oval to the screenshot, here, so to try and emphasize the numbers - they are most of the way towards the bottom of the screenshot):

Then, I need to select the Move Shape option from the Shape menu in Outfit Studio and enter those numbers there. And this almost works. But not quite:

So there is something else which I also will need to understand?

Edit: Oh, I see... it was staring me in the face... after moving to match the nif, I also needed to move it again using the "Old CBBE to Vanilla offset" preset amount. When I did that, it looked right (the outfit was clipped into the body, because it had not yet been adjusted, but it looked properly centered).