It ended up costing me around five dollars / box including shipping, which was a painfully large amount, but which should -- in theory! -- solve the problem for a long time. Assuming that the boxes actually fit. I was impressed when the company called to give me the tracking number for the shipment since their e-mail was misbehaving and they couldn't send it that way. And the boxes arrived on a timely basis.
Then they sat for a few days, because that's what it's like around here. But today, I popped them out of the shipping box and started folding.
And everything fits correctly. Of course, the boxes are so new that it will take a while for the top flaps to stay down of their own accord. :)