here is the statute from FL, Our statutes doesn't say running the pistol through a stolen is OK'd, but it doesn't say it can't be run either. (and running a pistol through a database has nothing to do with the 4th ammendment, improper seizure would, and in this case, it isn't) weapons are a special case (unlike drugs) Check out Terry and Ohio case
also, FYI, a traffic stop is a temporary detain. most are infractions, but still a lawful detainment.901.151 Stop and Frisk Law
(5) Whenever any law enforcement officer authorized to detain temporarily any person under the provisions of subsection (2) has probable cause to believe that any person whom the officer has temporarily detained, or is about to detain temporarily, is armed with a dangerous weapon and therefore offers a threat to the safety of the officer or any other person, the officer may search such person so temporarily detained only to the extent necessary to disclose, and for the purpose of disclosing, the presence of such weapon. If such a search discloses such a weapon or any evidence of a criminal offense it may be seized.
Hopefully this explains a little bit more.