Some broodies may sit 4 clutches in a year but some don't.
I think I understand what you are trying to achieve but think hatching under broody hens is unreliable and your venture may not be successful if you are working to very tight figures/percentages. Even really good broody hens are just that .... not hatching 'machines'.
If a hen decides not to sit she will not in my opinion change her mind if you lock her in a box. She will probably get quite distressed and break the eggs in an attempt to escape from her 'cell' as you put it.
I think your broody hens should have access to their food and water at all times ..... did I understand correctly that they would only go into the 'dining area' when you opened the door at a certain point in the day?
A broody usually likes to go into the run or at least well away from her nest to empty herself too. Often when she is lifted off for food and water but sometimes when she feels like it.
Think your broody hens would be denied certain 'freedoms' ..... if I have understood your proposed system correctly ..... that you are not allowed legally to deny.
Sorry if I have not understood properly ..... it is easy to do that on here sometimes.