I've been plagued by cold sores since I was little. Not one other person in my family ever got them.
I swear by Herpecin-L. It's in a tube like lipstick. Make sure there's nothing in it that can harm your baby if ingested but I know in my life at one time or another I got some in my mouth. I didn't get sick or anything.
Talk to the doctor and the pharmacist.
As bad as I've had cold sores, they fortunately never spread anywhere else which is saying something.
Don't worry about other people staring. I hate to tell you this, but your little one will likely get plenty of bumps and scrapes and scabs. It's just part of what happens.
My son's preschool photo recorded for history the lovely bruised, scabby bump he got on his forehead 2 days before.
I really hope your little one gets better soon. I've tried everything and I mean EVERYTHING. Herpecin-L soothes and doesn't sting, but they might have something that works better for your baby.
Best wishes!