I don’t think there is a straightforward answer to this question.
My first experience of such a thing is Paris, where most restaurants would serve ‘un menu’ at thus-and-such a price: so ‘menu à cinq francs’ (which shows how old I am). Typically, this would involve two or three courses with a carafe of wine: it was an all-in meal, as distinct from à la carte, and associated with a working day lunch. It was (and is) cheaper than the equivalent à la carte. Often it is called a ’menu prix fixe’ (fixed price menu.
In Britain and in America competition and diversification have left no standard expression. For one thing, it is not in either country the custom to eat a full meal at lunch on a working day. Many go to a restaurant for one course, which may be just a starter, like soup. Fast food restaurants and stands offer hamburger, hot dog, falafel wrap, pizza. That is not to mention pasties, pies and the great British sandwich and its gargantuan American cousin.
Many midday menus in the U.K. offer a fixed price option, with two or three courses, which may be a fixed price for two and three courses (often restricted to the less expensive items or with a surplus price for the chateaubriand steak or turbot). The one thing you will almost never find on a fixed price menu in the U.K. or the USA is wine included in the price, although many in the US and a few (usually American) in the U.K. offer free pop.
Moreover, the fixed-price menu is not confined to lunchtime. So mittagsmenü can have no equivalent in English. The solution in such cases is to import the German word into English as a loan word, like the Swedish smorgasbord The trouble with this is that there is unlikely to be any way such a German word could enter into the language. For all it’s diversity, the British landscape is almost barren of German restaurants.
So we are left with lunch menu and fixed price menu.