That would potentially solve the penalty issue, and while it may not be an ethical violation to take the client's funds in payment of their tax bill like it is for preparers to negotiate deposits of refunds, I think the client that should have paid would be suspicious and would want proof that the taxes were actually paid if their payment was to the preparer and not the IRS. Plus, the preparing firm shouldn't want its funds mixed up with those of clients in any way, and there's no way for correct client to pay harmed client directly without divulging harmed client's name because that would be an ethical violation. Either way, both clients will be unhappy no matter what the solution is.
Hopefully for the OP, the disgruntled client wasn't angry enough to contact IRS with a complaint of preparer error, and if they did, that the agency would see this as an innocent mistake, but we don't know that because we are supposed to be safeguarding information like this.