I doubt that either of the skin changes you've described are real reactions, in the sense of an allergy to the sunscreen (or anything else.) First of all, sunscreen allergies consist of scaly patches of skin that take days to go away. Runny noses aren't part of the package either. Second, light-skinned kinds (and people) get all sorts of minor variations in their skin appearance that mean little or nothing. Third, even if it was a minor irritant reaction of some sort, so what? Those generally mean little and go away on their own without stopping the offending agent. Plus, they're better than a sunburn. Bottom line: use any sunscreen for kids, make a note of anything you think is a reaction--photography is best--and see if it doesn't just go away by itself. Bring this information with you to the dermatologist.
Covering up is much better than sunscreen in a baby anyway, of course, so use the sunblock just where necessary.
Take care.
Dr. Rockoff