You need to provide more info. Why and by whom, was your child claimed. If it was the other parent, does that parent live with you?
A common error is when unmarried parents live together, If you and the other parent live together, only one of you can claim the child for any tax benefit. The interview is confusing (it's designed for divorced parents, who are allowed to split the child). The second parent should not enter the child, at all.
On the other hand, if you are the custodial parent, and are allowing the non custodial parent to claim the child this year, you should be aware that there is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the dependency to him. There's a good chance he claimed something he shouldn't have.
So do you suggest that I delete the second child's information even though on the return it says "non-dependent for EIC only"? Or should I file a paper return and risk the drama?
No, just the opposite. You are entitled to claim the EIC (if you are otherwise eligible), based on that child (I added something to my original reply earlier to explain that). "Non-dependent for EIC only" is the correct designation
The father, most likely, claimed something he's not allowed; probably EIC. Ask him to review his return. He can also not claim Head of Household or the dependent care credit. If so, he will have to file an amended return and you will have to mail in your return.
You could delete the child, for now, and e-file. Later, after the father has amended, you could file an amended return to add the child back for EIC.