The difference with Be Our Guest as a counter service (lunch service) is that you go through an entire "ordering room" before you even get to the dining room, and there is no ordering counter. So if you want a cup of water I think I'd ask at Gaston's tavern before I waited around long enough to ask at Be Our Guest, if the only thing I wanted was a cup of water.
I got a cup of water there but I was ordering from a cashier because I intended to pay in cash. If you are ordering a whole meal at the electronic kiosk, and you are not permitted to select a cup of water, you should not worry about it because as soon as you get into the dining room there will be cups for you to take and you can fill them with water from the drink dispensers. You dn't have to worry about them giving you a cup when you order. Just don't order a drink.