TexasCHLforum.com

A serial number on a firearm is not in "plain sight." In fact you've got to first seize the weapon and then look closely at it to even read the serial number. While the law allows an LEO to disarm a CHL holder for safety purposes, under the 4th Amendment the LEO still needs PC to "search" the weapon, which would include an investigation into its serial number (unless you are dumb enough to consent to it).

Again, it's analogous to having a laptop in "plain sight" on your passenger seat. The officer can only seize it and search it (which includes running the serial number) if he has PC to believe it is stolen or has been used in the commission of a crime. Again, unless you're dumb enough to give consent to a blanket search of your vehicle.

Of course, this is all pretty much a non-issue unless the search of your firearm does, in fact, result in criminal charges, in which case your attorney could move to suppress the evidence based on the search being illegal.

If an LEO ran the serial number of my gun after disarming me, I would have to question what gave him reasonable suspicion or probable cause to believe my gun was stolen/used in a crime/etc. and thus allowed him to legally run the serial number and "investigate" the status of my gun.