YTA
The dealbreaker for me is the child knowing they’re adopted. From that decision, I can reasonably assume one of two situations.
The child never knows who the birth mother is. The child will likely seek that information out on their own. Be prepared for confrontation when they find out (especially since it doesn’t seem like the sister is a bad enough influence to cut her out completely).
The child does know who the birth mother is, and the superfluous distinction between titles highlights a superiority complex over who is the better caretaker. Yikes.
Both choices seem rather self-serving from the perspective of the child. A couple alternatives: You can withhold that they’re adopted, and your scenario would fit the societal norm, or you can let the child know who the birth mother is and leave it up to them to decide. Both of these situations would be the most fair to all parties.