Ouch, that sounds painful. My friend's baby went on a nursing strike when he had a mouth blister from Hand,Foot & Mouth disease.
I'm no expert but I can't see why you couldn't give lysine to children - I mean it occurs naturally in many foods. I guess like other vitamins and minerals, just gotta be careful you don't use too much.
here's some info on dosages for children:http://www.acupuncturetoday.com/herbcentral/lysine.html
Aren't you worried about the cold sore spreading to your boob? That would be a major ouch. I got a damn cold sore right about I had my dd. I was terrified of spreading it to her, was too scared to even kiss her. I took that new oral anti viral med, Valtrex and it got rid of it before it could even break out properly, no sign of it after just 2 days.