[Pitchwife]

Not a native speaker either, but I think this could be understood either as "just as he held it" or "while he was still holding it". In any case I think you're on the right trace associating the dissolving of the blade with the growing light.

I'd like to add another angle though. The main purpose of the Morgul knife was not to wound the body, but to corrupt the stabbed person spiritually and turn them into a wraith. As Gandalf informs us (and Frodo) in Rivendell, the knife, or at least its blade, was supposed to have stayed in the wound - maybe indefinitely, maybe only until its victim had been fully subjugated by Sauron, we don't know. It may never have been meant to be reused.

Fortunately the blade missed Frodo's heart, but the splinter that broke off worked its way towards it and would eventually have accomplished its mission if left alone. So maybe it was natural for the blade to dissolve after it had fulfilled its purpose to leave a shard in Frodo's body, and it would eventually have done so anyway?