This isn't a given. My parents parted ways without any kind of fighting about who gets what. They decided between themselves how things would be divided and the court only signed off on their agreement. I don't know who decided how much child support would be, but you're not going to get away from child support no matter what you do.
There is also the issue that a Catholic man would not want or expect to "get out of" the responsibility (before God) of supporting his children.
Now he might want to support them under his own roof, rather than paying his wife (and some strange man) to start a entire 2nd household to accomplish this task, but I digress.
On some worldly, non-Catholic forums full of anti-feminists and such, some of the men talk as if they would love to get out of paying any child support whatsoever -- for their own children! That is too much.