I posted a query about the French place name on the American Name Society listserve and got the following reply from
Marc Picard:
According to Dauzat & Rostaing's
Dictionnaire étymologique des noms de lieux en France, Brienne (Seine-et-Loire), Brienne-la-Ville and Brienne-le-Château (Aube), and Brienne-sur-Aisne (Ardennes) may come from either Celtic briva 'bridge' + -onna or the personal name Briennos.
Unfortunately, there is no indication whether or not "Briennos" would have a similar Celtic origin to
Brian's.