You could have solved that problem the way I did and just had twins!!:)
Even with two, my 4 and just turned 6 year olds fight over who gets which one. I, too, consistently tell them they're not toys. I think it's more of a fight between the two of them then actually wanting the baby (I know they want to hold the baby but fight just to fight)
But, what has made my girls back off a lot is that I need their help, so I have asked them for lots of holding, or giving them pacies, or holding the bottle. Maybe with more baby care responsibilities, they won't be as aggressive. It worked for me.