Small, Hard Bump on Side of Nostril That Bleeds when I Scratch at It?

The lesion you are describing could be a benign angiofibroma, or a skin cancer called basal cell cancer. Both are common on the nose, and both can bleed. Very important to get a proper diagnosis by a skin expert, a dermatologist. Then a treatment plan can be developed for you. Don't treat a lesion unless you have a guaranteed diagnosis.

Hi and thanks for your question.  You did not include any photos so it not possible to comment about your particular case. However, there are many growths that can form on the skin anywhere on the body, and on the nostrils/nose. I would certainly have the specimen from the removal sent to a dermatopathology laboratory for an expert analysis to be sure there are no other concerns. From what you mention it would be important to be sure the bleeding is due simply to you scratching/irritating the area versus any underlying medical issue (i.e. a cancerous growth). 
Please seek out a board certified, fellowship trained Oculofacial Plastic Surgeon or other doctor with extensive expertise in cosmetic facial growth removal.  I have had patients travel from around the country (USA) to have me treat them for this kind of challenging removal - especially after they were turned away by dermatologists, general plastic surgeons, and family doctors.  I have removed thousands of moles from around people's faces throughout my career and each presents its own challenges, risks for scar, etc.
Best of luck in your search!
Damon B. Chandler, MD
Harvard-Penn Trained Oculofacial Plastic Surgeon

slow growing is a good sign. bleeding is not as good. I would have someone look at it like a dermatologist or facial plastic surgeon and it could need a biopsy. This should be covered by insurance.