This is only about a redefinition of some SI units such as the kilogram. With whatever changes in the SI units, the atomic mass will still be measured through mass spectrometry.
When mass spectrometry was invented after the 1910, they only go for greater improvements to get higher resolution. There is no published experiment that anyone made any independent verification if mass spectrometry is reliable. Our current analytical balance is accurate to 10¯⁵, enough to make chemical analysis of relative atomic mass of compounds composed of single isotopes of the elements. A simple example is sodium iodide; both elements exist as stable single isotope in nature.