My 3month old is having a hard time breathing because of his cough?

I would call your vet and just let them know what is going on! It likely could be normal, but any time a pet has trouble breathing, you need to provide appropriate veterinary care as soon as possible. Who knows, some vets will offer a no-charge recheck if you have just been there recently for an extended illness. They may be able to offer you an alternative cough suppressant that may work better for him and alleviate his breathing difficulty.