What can we do if the tax preparation service will not pay the extra fees?

Your question states that the tax returns were not filed on a timely basis. From that it appears that you have been charge late penalties and interest. If you can demonstrate to the IRS that the agent you employed caused the delay, the IRS may abate the late filing penalty but not if you did not provide information, documentation or cooperation to your preparer. Theoretically you should have paid enough in to cover your taxes before the return was due so a late payment penalty is not likely to be waived nor interest on the amount of tax due if you owe additional tax. If you can establish that your damages in the form of penalties were attributable to the preparer, you may claim an offset against the fees charged or seek damages through a lawsuit.

You would have to sue them, unless you signed an agreement requiring you to arbitrate with them. Before you do this, make sure you have read the agreement you signed, if any, with the tax preparer, and talk to the highest person you can within its organization. A judge may or may not agree with you, depending on the circumstances.

Sue them, if it is their fault. I do not understand why you would have interest and penalty, unless you owed. If you owed, and did not pay with the return, you would have interest and penalty anyway. Consult with competent counsel.

If you have a contract with the preparer that will govern the relationship. Is the amount you seek to recover sufficient to make it worth while hiring an attorney to review the contract and, possibly, seeking to recover the interest and penalties.