Infant cradle cap

My 2 month old has cradle cap in his head, behind his ears, & on his forehead. Sometimes the flakes fall on his eyelids, could that make them swollen? His left eyelid has been swollen for some time, it goes away and comes back. I took him to the doctor but they said it’s fine as long as it’s not any discharge coming out. I also have a dog (shih tzu), could he be allergic to him or is it to early to tell?