D.B.
Work with your pediatrician. If you don't like or trust your pedi, get another one. Ask if your child should see a pediatric ophthalmologist. A second opinion is never a bad idea, and any good pediatrician will refer you without judgment or anger.
Find out if massaging is a good idea. Rubbing in that area can increase inflammation, or can transfer bacteria from your hands to your infant's eyes. Please see a specialist before you take more of this into your own hands.
Do not, under any circumstances, put a home remedy in a child's eyes. I wouldn't put an over-the-counter product in an infant's eyes either. Even the most basic antibiotic (like erythromycin ointment) is a prescription product, and overuse or inappropriate use is a really bad idea. There are age restrictions on products for a reason.