The most recent SCOTUS case on point is probably UNITED STATES v. ARVIZU
and they firmly established that in the case of a vehicle "probable cause" has
been watered down to "reasonable suspicion" that a crime is being committed and
we all can only guess what a cop thinks is "reasonable". They are always a
little suspicious.
This is from the 2002 opinion, citing a 1975 case

"United States v. Brignoni-Ponce, 422 U. S. 873, 878 (1975), tilts in favor of
a standard less than probable cause in such cases, the Fourth Amendment is
satisfied if the officer's action is supported by reasonable suspicion to
believe that criminal activity "

It is always true that they can search, they have the gun. The only thing a
citizen can hope for is that a court tosses out any evidence they find. That
doesn't do anything for your privacy, inconvenience or damage to property.
Soverign immunity really makes it hard to sue them.