All of the above, but why is it critical that the groove is precisely centered for this application?
I assume you're trying to keep puzzle pieces from falling off the edge into jigsaw puzzle oblivion, from which it will never emerge. Therefore it is unnecessary to have it perfectly centered, in fact, having one side slightly deeper than the other is desirable.
My method would be similar, but I would get the test piece cut as well centered as possible, which would give the starting width, centered. Then I would cut all four pieces to that width. After that, make the adjustment require for a perfect fit on the test piece and once found, run all four pieces through. A perfectly centered groove would satisfy the picky, fussy part of me, but it would be unnecessariy perfectionist part of me.
Not any difference in technique, just in the sequence of cuts. Think about it, if you're using a full kerf, 1/8th inch wide blade, and you line up to the centerline of the frame pieces, you're going to be incredibly close, so the third pass, you are going to be taking off thousandths. That's pretty hard to do by bumping a fence to keep it perfectly centered. You would be making two passes, so your adjustment would have to be half the thousandths. Maybe for that you could blow on the fence to get it to move that miniscule amount.
A question, are the corners mitered? if so, you might want to use a spline to reinforce the corners, especially if the center board floatsk, is not glued down.