I would defenitely go see another dermatologist. A similar thing happened to me. I spotted something (a small pinkish looking bump) on my nose and after a few months I realized it wasn't going away, The first dermatologist I saw said it shouldn't be anything serious and told me not to worry about it. I saw a second dermatologist and he ended up performing a biopsy. A few days later I found out I had basal cell. Luckily mine wasn't something life-threatening like melanoma, but yours may be different so I would go see another derm. Good luck
It would appear worse than just a little spot. I don't think a dermatologist would look at, say, skin cancer and blow it off, it is not invisible after all. If you don't believe your doctor, go see a dermatologist or a specialist in skin cancer. In the meantime, keep the sunblock on your nose.