How should I respond to my cousin who lied to me and hurt my feelings, and should I completely cut ties with him?

Lying is one of the major sins. If someone lies to you and you know they are lying, it is obligatory to advise them, enjoin them to good, forbid them from evil, and warn them about the danger of lying, especially if the liar is a relative. You should not abandon them or curse them, because abandonment is not permissible for more than three days unless there is an overriding benefit, and cursing a Muslim is an act of deviance. Rather, it is better to forgive them and seek recompense from Allah Almighty.