"Their respective companies don't allow for anything but the supplemental withholding rate to apply."

Most typically, it's the Fed and State that have this requirement. The employer might have the option in NY (as with Fed) to include the bonus with regular wages, which will make the withholding even higher, typically.

This is exactly why your taxpayer should be managing their other withholdings as necessary. They can always make an Estimated payment, to keep prepayments on track. I always set my withholding to 0, for that reason.