My boyfriend doesn’t let me take my sister with me on a trip, but my cousin is lonely and pitiful on her own, and she keeps telling me to take her with me if I go out for fun.

You have to distinguish the nature. When a couple travels or does other things, it has nothing to do with the third party. The implementation of love is always a matter of two people. You'd better know that a couple's right to privacy absolutely excludes any third party, no matter how important that third person is to you. There are plenty of opportunities to take your cousin there, so why waste the two of you's time? Please think more about your boyfriend, consider his wish to be alone with you, and respect this wish. I believe that your boyfriend will also give you more opportunities to take your cousin out.