If you can make a small fenced in space within the greater run with a couple holes along the bottom JUST big enough for your birds to squeeze through, you could place the bird fodder inside and thus exclude the goats. Leave the bottom strand of wire on fence in tact to maintain the hole's shape. I don't know what your soil is like, but a few T-posts driven into the ground should hold the fence. Fencing goes on the *outside* of the posts, as goats will be pushing on it from the outside.
You need to make the excluder area deep enough that the goats can't pull some stunt like sticking a leg through to try and reach or tip the food container towards them. In other words, the bird feeding area needs to protrude out far enough from where you mean to place their feeders so that the goats cannot physically have any chance of reaching it.
Must be securely roofed one way or another to exclude highly motivated climbing goats.
I'd recommend making this up against a side of a shed or coop or shelter, stapling the fencing to it. Saves fencing and that way you can mount a feeder up against the thing so there's no chance of the goats reaching it, period. It can't shift around, be knocked over and rolled around, etc.
We can manipulate stuff like this to suit our favor, but it comes at the price of having to be diligent and think of everything beforehand, otherwise it could cost you your goats.
...All this would have been much easier to describe with a 30 second drawing, but these fakephones suck. Lol. Let me know if it makes sense.
I've fabricated similar things so my chicken chicks could eat certain foods I meant for them to get, while their fat, surly adult flock mates couldn't gorge themselves on the babies' food. Babies were able to come and go freely through the small holes.
