While I haven't had any experience with this particular question, I breastfed my daughter for 2 1/2 years and know completely the struggle you face trying to keep your body and your milk as healthy as possible for child.
I did a keyword search for "L-lysine contraindications" and found a website with detailed information. I thought it was a strange place to post such info, on a dentists website, but I believe the information will help you.
Good luck with recovering your health and kudos for the "extended" breastfeeding!!!