Here is my situation. I keep my chickens in a run and they dont mix with the goats however, the following is what I have set up for feeding my geese, who live with the goats. It works well and it would probably work with your chickens possibly. You could always try it.
We have a old metal auto water for cows..basically, it is a big metal box, say about 4'L x 3'H x 3' Deep. it has a hole on one side and a door on the other. Top is solid. We put this on the opposite side of the fence we use for our goats, and our fencing is re-mesh or cement wire , has 4' squares and is 5 ft high. We placed a wooden pallet against the fence to keep the goats from sticking their head through to the feed.
The geese can get their heads through to eat from the plastic bin inside, but the goats can't.
What I suggest, is to build a large box (make is strong so the goats can't destroy it by jumping on it(cause they will)if you are going to put it in the enclosure with them. solid top, door on one side so you can refill the feeder and a open side on the other for the birds to eat out of. you can add wooden slats or dowels nailed to the side they feed out of spaced far enough apart so that the birds can get their neck through, but the goats can't. Another idea is to build the box like I said, but put the feeder inside the box at one end on the box and a chicken size door at the other far end. If you make this deep enough, the chickens can enter to eat, but even if the goats can get their neck into this box, they still wont be able to reach the feed. So make it longer than the goats neck! LMAO. Oh, and you might want to fix it some how to the ground so the goats can't tip it over either.