That とき is the verb 特 (とく), which has several meanings including “to dismiss”, in its ます stem form.
As you may or may not know, it is common in Japanese to combine verbs by using the ます stem of the first verb and then conjugating the second verb normally. In this case, なさい also needs the ます stem of the verb it is attached to, so both verbs ended up being in their ます stem.
You can think of this way of combining verbs as a formal version of the て form, thus this becomes “to leave behind and dismiss”. Obviously that’s more of a literal translation, but you get the idea since you already figured out what was being said.
HTH!