Just call me Erik
I don't think there would be anything particularly fun about that. It would be a lot of work, and I don't think it would play well. It would just be more random, and it would be harder to complete the items.
Not to mention each round would last forever.
Why not try making every menu available, play with the entire deck, and see what happens? At least then if it's terrible, you didn't waste time and effort on a new menu.
EDIT: You would need three dice: one for the menu, one for the section, one for the item. I can't reember how many menus there are, but I know it's 4, 6, or 8, so the menu die would be something easily found.