Did your sister refuse the money? Or, did your daughter never offer? There is a huge difference. If we assume she did not even offer to pay for something then two things are going on here.
One, she has a tight grip on that money for selfish reasons. Two, she did not do as she was told by you with the money, which means she basically stole the money. She was to be the intermediary to get the money to the right place as you dictated, and instead she swiped it.
I’d take the money back, tell her the money was given to her to pay for her aunt’s meal, and not for her to keep.
I’d point out her aunt has been spending money to take care of her, and that the appropriate thing to do is to reciprocate.
If I’m understanding the situation right.
Other money, money that is gifted to her, I would focus on saving more than anything. Teaching her to be giving I would do by example more than anything. Let her see you pick up the check, give to charity, and help others in other ways. Not that she can’t use some of her money to be giving. One idea is for her to choose a gift for a child for Christmas with one of the organizations that give gifts to the less fortunate with her money after saving all year.
I think your original idea was just fine. Giving her money so she could treat her aunt, or pay for something herself while with her aunt. If your sister possibly told her to keep her money, then next time let your sister in on the exercise so she accepts it from her niece.