This is an intelligent and detailed question and though welcome because supplements has not hitherto been an area of my expertise it will take a while to answer in the detail it deserves. To start with, though...

Earl Mindell writes in his "Supplement Bible":
"... For more than a decade, lysine has been prescribed by natural healers and mainstream physicians to treat cold sores and tother symptoms caused by the herpes simplex-1 virus... (it) Helps alleviate symptoms of herpes infection and helps prevent recurrence...
...Lysine is available in capsules and tablets. to prevent a recurrence of herpes, take 500 to 1000 mg. daily."

Supposedly lysine has been observed to inhibit the growth of Herpes simplex I. H. simplex lives in the nerves in the Trigeminal nerve ganglion. When one is under stress and the immune system is therefore somewhat disabled, the Herpes virus migrates down the nerve axons to the lips, creates a sore, and waits to get kissed over to another "host". It would make sense therefore to use the lysine both "topically", as an ointment on the sore, and "systemically" internally to effect the whole system. (It might help to take ginkgo too as ginkgo increases circulation to the nervous system.)

"Natural form of the stuff"? Lysine is an amino acid: one of the 21 amino acids that make up natural proteins. About half of these 21 amino acids can be made by the human body from other amino acids, and about cannot be synthesized by humans and must be derived nutritionally from protein intake. these are the "essential amino acids". Lysine is an essential amino acid.
(Incidently, corn, maize, lacks lysine and if your diet is totally centered on maize you can suffer from a deficiency disease called "pellagra".)
Most proteins are made up of a melange of most of the amino acids. As of this time I am not aware of a source with a therapeutically useful lysine content. (One of the source I chased gave as nutritional sources, "beans, dairy, eggs, fish, meat..." - that is to say, high protein foods - and "...potatoes..." which is interesting because...) I've just received some feedback that, "Potatoes are the best natural source of lysine (with the skin on)." Nonetheless, I think that by and large you are stuck with supplements.

In my practice I would approach the treatment of cold sores through: