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He is going to need another in person exam.
This might be due to a nasal infection, a tumor, an abscessed tooth, or if he goes out or any rodent or insect poison was used in the home it could be from a blood thinning issue.
If anyone in the home is on heparin and the cat might have gotten a capsule that can cause bleeding.
So can autoimmune disorders.
If he might have swallowed anything sharp or string or ribbon this could be bleeding from the esophagus that he is coughing up too.
You can read about rodent ulcers here
Epistaxis or nasal bleeding can be caused by a number of things too. You can red more about that here
I think a more in depth look at what is going on is needed and if your vet doesn't feel they can do that you might ask to be referred to a small animal internal medicine specialist or to a feline specialist.
Hope this helps you!