You say that you have a 4-year-old child. You also state that you picked MFS only based upon what "seemed to give" the biggest total return. I would suggest making test returns to "know" what is best. A joint return is one return that should be tested.
I don't know the questions that the IRS would ask if you choose to amend to try to switch dependents. You might provoke some questions, and you need to be sure that you are truthful and are not committing any fraud to manipulate refunds. As you indicated, the Social Security representative indicated that you are unlikely to get any money returned because you owed them the money. And borrowing money to pay the IRS because your refund was used to pay money that you owed might be an endless trap.
I cannot make a specific recommendation for your question. But here is some information that I hope will be useful. Because the child was with you more nights, then the dependency should be yours. If nights are same, then using strict dependency tie-breaker rules, the dependency would normally be applied to the person with the higher AGI since that person would usually tend to maximize the potential benefit from the child tax credit. A portion of the child tax credit is non-refundable. The non-refundable part can only be used to "pay taxes", while the Additional Child Tax Credit is a refundable credit where money is refunded even if no more tax is owed.
Before going forward, you should look at what would happen if you filed jointly. Filing MFS eliminates the possibility of the Earned Income Tax Credit (EITC). MFS also likely eliminates getting any day care credits if you use day care to permit you and your wife to go to work.
Please run the MFJ scenario. If your AGI/earned income is in the neighborhood of $45,000 or less, then the MFJ filing status might result in an EITC . You cannot get EITC if you use MFS.
If your spouse is not legally responsible for your debts, then MFJ can also be possible using the process for Injured Spouse. I am not an expert in doing those tax returns, but here is a link with some information: