Wouldn't a Red-Black tree fix up after insertion mess up the BST ordering?
A correct implementation of red-black trees will maintain the BST ordering property. Whenever you add a node to the tree or delete a node from the tree you very carefully maintain this property. If you have to modify the structure of the tree then you do so in such a way that maintains the BST property.

I suggest you go carefully through the insertion and deletion pseudocode and verify that both operations maintain the BST property.