"There is no problem. Don’t register with the parish and don’t initiate contact with the pastor. You mind your business and he’ll mind his."
 
Parishes have thousands of families in them nowadays.  The fact that a priest has time to nose into the living arrangements of one family that hasn't asked him to suggests that he has too much time on his hands.  I provided some helpful suggestions for how such a man could better spend his time.  

"Children are routinely denied baptism if there is not a sound belief that they will be raised as Catholics."

It seems like the only thing that the priests' focus on is the parents' "irregular" relationship.  Otherwise, most children would be denied Baptism.  

And it certainly is easier for a child to get the rest of his/ her Sacraments, marry in the Catholic Church, etc. if the Baptism is on record.