Honestly, my instinct in this situation would be to explicitly ask the sister if she wants LW to correct the relatives when they use Sister's middle name. Obviously she shouldn't have to settle for that "compromise" (ugh!), but by telling LW she "thinks it's good enough," she may mean that pushing back on it would cause worse problems for her, and that may be true even if LW rather than Sister is the one pushing back. LW should make it extremely clear that they're willing to get in a fight with the awful relatives about it, but it should ultimately be Sister's call.