If we can specify our child custody arrangement, does the court have to approve it?

No, the court does not. As noted, separation agreements will be scrutinized by the judge. The "best interests of the child(ren)" rule will always be at the forefront. The judge will not take it for granted that the decisions you and your spouse may have agreed upon are best for the child(ren). He or she will ask you and your spouse a lot of questions in order to substantiate the request.