What does a 3-year-old have in common with a 14-year-old? Would you want to watch the same video over and over again? Do you want your 3-year-old involved in teenage activities? Your insistence on forcing them together is harmful for both girls. My experience tells me that if you let the girls naturally share family activities -- meals, vacation time -- the "meanness" will disappear.
Because of the age differences it will be very difficult, if not impossible, for the girls to relate to each other in the close relationship. But they can develop a truly loving sisterly relationship if you let them develop it naturally and understand that each is at a different developmental stage. You say they won't have to share a room after you move. Stick to that promise and if at all possible separate them now. Your teen doesn't have enough space if she is sharing a room with a three-year-old.