Why can’t I use solubility product for normally soluble compounds?
In addition to Mithoron's and Poutnik's comments, the solubility product may not be relevant to your solution chemistry anyway.  Solubility products enter into solution equilibria when the precipitates are present, as in this answer where relative solubility products predict whether mostly magnesium or mostly iron will be dissolved in water when their hydroxides are present.  With readily soluble materials, you likely do not get the precipitate so solubility products, even if you can compute then on an activity basis, have no bearing on the equilibrium.