I wouldn't want the other mom to send food over with her kid unless I knew that it would be something unexciting. If she sent for her kid brownies, chips, lunchable meal with candy, etc, then it wouldn't be fair to my kid. Just something to consider before you ask her to send food.
I would never force a child who was not my own to eat, with any degree of coercion, but I do insist my own child try things. He's laid back and a naturally easy going eater, so I realize how easy I have it (there's no actual force involved, of course). I honestly do believe most picky eaters are made, not born, and I see a lot of parents in my community who I think don't challenge their children enough with new foods, especially vegetables.