I agree that finding a free service to amend that return is going to be a challenge this time of the year and time is of the essence.
However, though the mistake was glaring one —HOH vs MFJ, it’s a change of status. According to OP, his mother had no income. It should be a simple change for the tax preparer who has the return in his system. He would just do the 1040x based on already done return with the only change being that of filing status. It would change the standard deduction if used and possible other credits, but That would be the quickest way to do it. Any one else would have to pretty much redo the return in their system or do by hand. Without knowing complexity of the return, can’t say how time consuming this could be