If your black background is not the stage color and instead an image or quad, it will need to be added outside of the sprite you are fading, or else it will simply fade with your other objects. I assume this isn't the issue though. Besides that, your idea should work just fine.
To reuse the tween, just make it an instance variable.
What do you mean by "changing its children when its alpha is 0" and "I've tried setting delayed calls using the juggler"? You should not need to modify the sprite's children or use delayed calls. Fading the sprite should fade all it's children.
The tween code would look like this:
_fadeTween = [SPTween tweenWithTarget:sprite time:1.0f];
Provide your code if you would like additional help.