Routing belongs in the menu: every item names the station that makes it, and a mixed order splits itself into one ticket per station.
When staff ring an order in at the register, something has to decide which station makes each line of it. That decision belongs in the menu: every item on it is assigned to one station, and a ticket goes wherever the item says it goes. A pint is a bar item, a burger is a grill item, a flat white is a coffee counter item. None of it belongs to the person on the register, who should never have to know which door the food comes out of, and none of it belongs to the printer, which prints whatever it is handed.
Keeping the rule in the menu is what lets a mixed order split itself. One order with two drinks and two plates becomes two tickets - the drinks at the bar, the plates at the grill - and each lists only the lines that station is responsible for, with the order number and the guest name on both so the halves meet again at the hatch. Neither side reads past its own work.
Some items are sold everywhere. Water, crisps and a coffee are wanted at whichever counter the guest is standing at, and the answer is not a clever rule but a plain one: each counter holds its own copy of those items, pointed at itself. The duplication in the menu is the price of never having a bottle of water print across the site from the person who asked for it.
Some items move, and some are made in one place and handed over in another. A station closes early, a second grill opens for the evening, a tray is prepared in a back kitchen and collected at a front counter. Reassigning an item should be one change in the menu that takes effect on the next order, with no printer touched and nothing reinstalled at the counter. A handover is simply two stations and two tickets: one to make it, one to say whose it is.
Set the printers up the day before and ring real orders at them rather than reading the menu back to yourself. Ring one item for each station and confirm the paper comes out in the right place. Ring a mixed order across three stations and check each ticket carries only its own lines under the same order number. Ring an item with modifiers and read the ticket from arm's length, the way it will actually be read.
Then pull the paper out of one station mid-order, finish the order, reload, and watch the missing ticket arrive on its own without anybody retyping it. If you also let guests order from a code at the counter, ring one that way too and confirm it routes to the same station as the register - the same routing table, not a second one to keep in step.