Unfortunately, there won't be anything you could do at home with a respiratory problem connected to an injury. Cat claws/teeth are full of nasty bacteria, which is why you get slight swelling, redness and pain after scratches/bites. Worst case scenario, it could be an internal abscess from an infection that's pressing against the lung or something like that. Best case scenario is just a narrow nasal cavity, my cat has that and she's got no issues with it, but makes a lot of weird sounds because of it (she has been seen by a vet about it).
I don't know if you have, but I suggest calling the vet and asking if they do have any experience with exotic animals - sometimes they have a lot, it's just not advertised as "their field".