catterygirl, surely a much less drastic solution lay in yours and your DHs hands, have a second child, but treat them both equally.
The root of sibling rivalry is either preferential treatment by parents or the encouragement of competititon between siblings, also by parents. There will be the occasional exception, but it is only occasional.
I do know families where sibling rivalry has almost destroyed it. Read GN to find more, but in most families siblings get on well and are a loving support to each other.