"The state amounts of income add to the total income"

If these are all from the same employer, you don't expect the State amounts to add together to equal Fed. That's because the Feds use the total. But various States have different rules. Some will have you report the amount earned only while in that State. Some require you to report all earnings (the same as the Fed amount) and then they allow a credit back from tax paid to other States. Some States expect there to be an allocation, such as by % or number of days worked in that State. And some States will tax the amount from the employer no matter where it was earned, because that is considered the Source State, even in the employee only worked telework or remote.

It's too late now for payroll to do 2022 manual withholding. Any withholding that was in their State tax tables has been done for the year. Your client likely didn't hit the tax tables for withholding. If they know that they will owe, then they need to update their records for 2023, and perhaps make a voluntary 1st quarter estimated payment to that State, because here we are mid-April.