I think I like your constructor change idea, with a small twist:
Color(value=<sentinel>, name=<sentinel>, default=<sentinal>)
This would make it possible to search for an enum by value or by name, and also specify a default return value (raising an exception if the default is not set and a member cannot be found).
So specifically this would allow (hope my shorthand is clear):
Color['RED'] --> Color.RED or raises
Color(1) -> Color.RED or raises
Color(1, default=None) -> Color.RED or None
Color(name='RED', default=None) -> Color.RED or None