You talk of “giving a gift” to the one who hurts us. Yet, in some cultures, it is considered rude or disrespectful to hand out gifts. It is seen as a sign of superiority. So, might it be best to refrain from “giving a gift” in such cultures?
In such cultures, as you say, it is best to give the gift in ways that respect the norms of the culture. One need not give a gift within a box all wrapped up in gift-wrap and a bow. One can be more subtle about it: a smile, paying respectful attention to the other, not speaking badly to other people about the one who hurt you. A gift is a generous and often unexpected kindness which can be done tastefully by knowing the norms of a given culture.