If you aren't seeing any improvements with the lysine after a few weeks at the higher dosage, it's probably not helping. If it helps at all, I'd be tempted to keep taking it (maybe at 2000 - 3000mg).
There is always the distinct possibility that this is (at least in some cases) a form of eczema/dermatitis. Lots of things can trigger/exacerbate it including stress & environment.
Has anyone tried serious stress management (regular exercise/meditation/yoga) and/or anti-anxiety medication? Just wondering.