In this case, the bleeding could well be a result of long term steroid use. Was it prednisone (or it's active form prednisolone) that Dopey is taking? Two years on steroids is a very long time and you should maybe talk to your vet about weaning him off and trying a different medication.
Prednisone can cause weight gain, osteoporosis, easy bruising
, and endocrine problems when used long term. It could be that the blood vessels in the nose have become weak and the sneezing caused them to break.
Nasal cancer is, unfortunately, very hard to diagnose unless a biopsy can be obtained. However, at this point, I'd be more suspicious of side effects from the steroids than nasal cancer.
Good luck dopey.
Please keep us posted.