I don't understand why you would pay out more money. As you say the correct amount has been paid out.
Aren't there really two problems here:
1) The 1099-R was wrong in terms of how much it said was taxable. So isn't the solution to that to file a corrected 1099-R? The person would need to file an amended 1040 to get back the excess taxes paid. (If the 1099-R was issued to long ago that they can't amend I am not sure I wouldn't let a sleeping dog lie as they say)
2) The amount of basis on the recordkeeping system needs to be changed to reflect the amount of the correct after-tax basis that should have been paid. I guess if the after-tax money is in a separate daily account you might need to do a transfer to get things in the right place. I would have to think about that more.
Once again I can't for the life of me see why you would send more money from the plan. That seems to make the mistake worse as you now have paid more money that was needed to pay the distributions. That really seems like you would be going in the wrong direction.