It's amazing what you, as parents, can get away with regarding your kids. I always told my kids that the end pieces were the "special pieces" because they had all that extra flavor. They had no problem eating sandwiches made with the end pieces, and they always ate crusts and everything because of the extra flavor in the crusts. Anything I thought might pose a potential issue, I'd tell them ahead of time how it was "special". I don't know why that was the magic word, but it worked.
My wife's mom would just invert an end piece, so at first glance it looked like a regular piece. And in her family, you just ate what mom gave you. There was no argument.