Healthy provisions I don't think have to cost so much as people like to imagine, but it means parents need to get on board with school decisions and what is going on there. Forcing children into only being allowed school food (especially if it has to be bought) is a pretty bad practice too, because so much of the food is going to be wasted if it's overly health focused.
A true cafeteria system would make more sense. You get a main entree for a specific price, and the addition of whatever else at an additional cost. It would help to manage waste and cost a bit since children could skip over something they do not want and/or substitute it for something else.
Also, assuming the Japs still practice it this way, use the classrooms as the place for eating and/or recess, where the teachers can keep some eye on them. They go to the "cafeteria" (where food is prepared and served) and return to the classroom to sit down. Hallway sides of the classroom should be made of glass windows so tabs can be kept on students without having to be in the rooms themselves, but students would be able to freely move about around the classrooms and hallway if they choose. So essentially it would be a lunch and socialize period with a possible outside recess tacked on as well. The Japanese also I know have in some schools the students take part in janitorial duties, especially classroom upkeep which would reduce the time and cost of having a dedicated janitor, whiling instilling some sort of personal responsibility and pride in the students in keeping their classrooms tidy. While I wouldn't want the Japanese obsession of pressure education, the practice of making the students involved with more than just their schoolwork could be an important part of their personal development by teaching personal and group responsibility. Surely, school days might be a bit longer, or a need to move to a 6 day school week might be needed (which I kind of wish would happen), but there are plenty of things the US could implement in some moderate form here that might fix some of the stupid BS that exists already or is in the planning stages.