Hi, I also have a bully goat , my Dozer eats everything and anything , he has eaten socks twice, grass, pinecones, rocks, a dish cloth, and who knows what else is in his stomach. 2 nights ago he ate a part of a plastic grocery shopping bag, my husband put the bag down near his cage, and he managed to pull it to his crate and I heard rustling, I jumped up to check and he had already eaten the pieces around the handles. Now we watch and wait and play CSI with his poop again.
Everything that he's swallowed he's managed to pass with no problems. The first two times we took him to the vet immediately and had X-rays done. She gave him a laxative and fibre food to help move the bowels, and within a week the objects and sock passed.
You are probably okay because the object is small, round and smooth. If it was a pointy or sharp object you would have to worry about it puncturing or cutting him on the inside, but since its smooth, it most likely will pass through.
The signs you want to look for are vomiting, diarrhea, or drooling which are signs of an obstruction. Just keep an eye on him, and check his poop everyday until it passes. This could take a week or more. If he seems to be choking, coughing, vomiting, or has excessive diarrhea, I say excessive because Dozer always has a day of either diarrhea, or soft poops before he passes something, so this is normal, but if he has excessive diarrhea or vomiting, then take him to the vet.
Good luck, and I hope he passes the water cap soon. Keep us posted