The problem is that the State Department can't enforce laws or kidnap children on foreign soil, even if US law recognizes the illegal taking of the child. They don't have any jurisdiction to do so. What they CAN do is set up border blocks preventing removal of children from the country under certain circumstances, but when parties are still married, the contract of marriage is more or less a legal permission to move the child. That's why the advice you're going to see over and over is: if the marriage isn't working to the point where who has the children with them is an issue, then divorce is the way to void all of the inherent permissions that go along with marriage and allow a court to set forth very specific instructions and permissions in PLACE of the inherent ones in effect now. Filing for divorce doesn't mean you can't reconcile with your husband at some time in the future, but so long as you choose to stay his legal spouse, his rights to the children will be difficult if not impossible to curtail.
Again, if you feel you must leave the birth certificate blank, then by all means buy yourself some time to deal with the issues at hand and do so. But if you fraudulently put some OTHER person's name, you are going to be facing a legal situation that will turn around and bite you (and could have a very negative impact on your ability to retain the child in the long term). You will only HURT your efforts to keep the children in the long run by doing so.

