It was Brightan’s purse.
I think he put it in her purse as a misdirection.
I also think he may have publicly flirted with her as a misdirection.
My theory is that he wanted Clint and Meredith to think he left the bar with Brightan, and he wanted his cell phone to support that idea.
Investigators spent a lot of time and effort tracking his phone pings for the entire month of April.
If you haven’t already, I encourage you to watch Ed of the Unfound podcast break the video down frame-by-frame. It appears Brian reaches into his pocket, looks cautiously over at the cop standing nearby, turns with his back toward the cop (you can see a silver object in the same hand that was just in his pocket), he looks down at that object, then reaches behind Brightan very quickly with a dropping motion toward her purse, which is a handbag hanging open from her forearm. When his hand returns to his side, the silver object is gone.
Remember he disappeared from sight & was unreachable several minutes later.
MOO