You talk about giving a gift to the one who offended you, but in some cultures it is considered rude or disrespectful to hand out gifts. So, it seems to me that this is not a good idea.
You raise a good point about how we are to be merciful or loving toward those who were not merciful or loving to us. Generosity is a virtue that would seem to be universal. It is in how this is demonstrated that is at issue here, not whether or not to ever exercise generosity. So, with that clarification, I would say that one should be sensitive to the cultural nuances of what you have in mind as your gift. If handing out gifts is seen as showing off or condescending to those receiving the gifts, then it is best to be quiet and private in the giving. One need not give a physical gift, such as perfume in a wrapped box, for example. One, instead, can give a smile, or respectful attention, or a good word about the person to other family members.
