Don't buy into the Red Herring arguments about stability, back and fourth, etc. It took both the produce the child, both to raise the child, both to contribute emotionally and monetarily-but then they split up and it's not healthy for both to be equally involved?
If you want to fight it, spend ass loads of money then go for it. But I think the time and money could be better spent-say preparing the child for higher education and putting the money aside to pay for it?
Many times the resistance is simply a control issue-if everyone has an equal stake then there is far less to fight about and less for one to try to control the other. Think about it.