The following method is imprecise but repeating with no Na salt, that is just adding water, and measuring change gives a measurement error estimate. The method is also theoretically scalable.
Start with a long thin tube containing a Potassium salt (like KClO3), that has been cooled to a specified temperature and precipitated out. By standing on end, measure the depth of the potassium precipitate. Pass your Na salt into the tube after warming the solution. Cool back to the specified temperature, and measure the new depth of the K precipitate. Loss in height relates to a possible replacement reaction with a more soluble Na salt. For example, NaCl impurity would form NaClO3 and KCl, both more soluble than KClO3.
The essential concept is to take advantage of the higher solubility of Na salts as a path to a metric.
The problem would be much easier if we knew the composition of the impurity like Sodium Sulfide. Then, a Ag based test, highly sensitive to the presence of Sulfur, could be devised.