Hi W.,
Neosporin contains petroleum products which will dry the nose out further and antibacterials which in absence of any infection, he does not need.
What the nose needs is moisture and a chance to heal. I would 1st make sure he is well hydrated. Plenty of fresh water each day and canned food hydrates as well. Then I would put fresh aloe vera on the nose from a freshly cut leaf. The plants are inexpensive and one will last a lifetime as they grow well under most contitions. I keep mine is a pot for such uses. Just cut the leaf open and scoop out some gel and place it on his nose. Aloe has healing properties for exactly the tissue you are referring to epithelial cells of the skin and the lining the nose, and digestive tract.
Place the gel on when he is relaxed and then give him something to distract him like a bone or cookie. Watch for 10-15 muinutes to discourage licking. The aloe won't hurt him if he eats it (unlike eating neosporin) but you want to let it stay on the nose a few miuntes to be absorbed.
If your dog's nose is pink, instead of black (which has more melanocytes which protect from cancer from the sun) after it heals, you might try putting on sunscreen made for the face.
Good luck!
L.