Toothpaste is not considered assur by everyone, but the minhag is to be machmir like R’ Moshe, the Tzitz Eliezer, and others. There are more reasons to be machmir by ChapStick (it’s thicker, and stays in the skin, whereas toothpaste is thinner, and washed away immediately). Lipstick has the additional problem of tzovea.
But my point wasn’t that the issur is universally held (although that is the mainstream psak). My point was that you made up a heter of kilachar yad. ChapStick comes in a stick, is meant to be applied from the stick, and that is in no way shape or form kilachar yad. Note that the OP didn’t say what type of lip balm was used or how it was applied; you just assumed, for no reason, that it was done kilachar yad. (Even if one did apply it kilachar yad it would still be assur in the vast majority of cases, BTW.)