No one will be arrested upon entry without an arrest warrant. The ex-husband might have complained, might have even made a report or two to ICE or others, but unless a warrant has been issued, no arrest is going to occur.
If the ex wants to claim paternity, he can seek a court order. If he merely wants to get his ex-wife into trouble, that might be problematic since mom is still mom and she is alleging that the boyfriend/fiance is the child's bio-father, so the intent required for a fraud/forgery case might be difficult even if technically correct.
I cannot recall from my own childrens' births whether we had to fill this out or sign anything, but I can't imagine anyone making a huge issue about this under the circumstances. And if the ex-husband alleges to have spoken to Homeland Security and the National Guard about this, then he is full of hooey as they wouldn't address these matters at all.