Looks like Dad needs a time out to cool down, stop the power struggle. If the child is that much of a wanderer, that difficult to shop with, then don’t bring her into a big supermarket full of distractions. Or bring a second person to hold her hand. I guess using a thin braided leash with harness is not allowed in today’s world, at least not with a child that old.
Not always are the suggestions you offer viable, however, putting her in the cart would have been my first step. If that failed I would take the matter else where and delayed my shopping. Public discipline will most assuredly get others involved, and I do not want others involved. That is especially true when those others, involve members of a government agency of one form or another ie. police or child protective services.