I'm sorry, but let me make sure I understand this point correctly.
You have a person who:
A) The bank staff felt was enough of a threat (wrong on their part) to push the panic buttons, and
B) The bank staff (to the best of their knowledge, though incorrectly) identified as a suspect in several Tacoma-area bank robberies
and you feel the best approach would be to stop him and talk to him? That's what is called, in scientific circles, "silly".
I'd like to see your point of view, truly, I would. My POV, however, is shaded by personal experience and training. I once blatantly disregarded my training and drew upon my personal experience when contacting (in a "stop and talk" manner) a person who was said to have been threatening someone with a knife. Seeing as it was a person who I'd previously contacted multiple times, sans incident, and being in a crowded area, I chose to disregard proper procedure and merely "socially contact" this person.
61 stitches and a few months of physical therapy to regain the use of my left hand later, I'd learned the hard way. I hope and pray you never have to make that call.