An easier solution, from the standpoint of future litigation, would be to ban the use of such anchor babies to allow the parents and other family members to remain in the US. Just deport the parents regardless of the presence of such an anchor baby or anchor babies. Let them decide whether to take the kid(s) home with them or not.
That would probably be achievable by Executive order, but at the most it would also require a change in the law. But it would be a law much less subject to challenge on Constitutional grounds.
I tend to disagree that a mere law can redefine the terms used in the Constitution. That's a very dangerous path to go down.
This power is, as I say, right there in the text of the 14th Amendment.
John / Billybob