yes, some additional screening - i would put them through "field tests" ... real life relatively common scenarios. several "traffic' stops..one is a bad situation : see how the applicant handles all three. put him in a hostage situation...a suicide by cop situation. possible domestic, out of control tweaker type.. they wouldn't know what they would be looking at . just random mini situations..get BP, heart rate stats, time reactions, do they defuse a situation or try to heavy hand command it immediately? how do they handle a potentially dangerous mentally  ill person who's going off? stress them out...see the results.
 i don't  know ..but written test,  interviews and shooting range isn't enough.