The pair interaction between two charged colloidal particles, in the presence of a polyelectrolyte as well as simple salt, is analyzed theoretically. Of particular interest is the way in which such a combination of salts can be used to induce a strong, long-range attraction, with at most a minor free energy barrier. We show that the nature of the simple salt is highly relevant, i.e., 2:1, 1:1, and 1:2 salts generate quite different particle interaction free energies at the same overall ionic strength. We adopt several different theoretical levels of description. Defining simulations at the primitive model level with explicit simple salt as our reference, we invoke stepwise coarse-graining with careful evaluations of each approximation. Representing monovalent simple ions by the ionic screening they generate is one such simplification. In order to proceed further, with additional computational savings, we also develop a correlation-corrected classical density functional theory. We analyze the performance of this theory with explicit spherical particles as well as in a flat surface geometry, utilizing Derjaguin's approximation. The calculations are particularly fast in the latter case, facilitating computational savings of many (typically 5-7) orders of magnitude, compared to corresponding simulations with explicit salt. Yet, the predictions are remarkably accurate, and considering the crudeness of the model itself, the density functional theory is a very attractive alternative to simulations.