Thanks for the suggestions so far guys :)
I’m already wetting most of the hay, and he always dips it into his water himself too
Sounds like allergies! Unfortunately, there's not a lot you can do about that from what I know. One of my boys has the same sort of cough after he goes running around, and we have him on SmartBreathe pellets. They seem to be helping.
Make sure he gets lots of water and lots of salt and you’ll see him stop copying in a few weeks