If anyone wants to read about the legal issues, I found an article which seems to go through the issues pretty well.
See Sarah E. Waldeck, Using Male Circumcision to Understand Social Norms as Multipliers, 72 U. Cin. L. Rev. 455, 515 (2003)
The general rule is that “the Free Exercise Clause permits a law that burdens religion as long as the law is neutral, generally applicable, and is not passed to ban behavior solely because of its religious motivation.” The leading case on this is Oregon v. Smith, 494 U.S. 872 (1990).
However, the Court also said there in dicta that when another constitutional claim besides for free exercise is also implicated, then there is an even stricter level of review. That would likely apply here, because of the due process claim.
Also, there might be an issue here since it would practically make it impossible to live as a Jew or Moslem, like Pierce v. Society of Sisters, 268 U.S. 510 (1925), which struck down a law prohibiting parochial schools. Apparently, this makes it much harder, although I am not sure how.
This article concludes that a law requiring anesthesia would probably stand, but not a law banning bris altogether.