The answer came to me, of course, a few minutes ago: Clothes Make the Man by Henri Duvernois; this site seems to have a copy (or, at least, a detailed synopsis) of the story.
It turns out they weren't robbing a bank but a house, and the main character doesn't turn himself in so much as he arrests the other two.
Along the way, another officer salutes him on his "patrol", he helps a little old lady across a street, and gets spat on by an angry drunk; the latter he dragged "off down the street".
Our main character, Tango, is treated poorly by the boss (unnamed in the link) both before and after the job. When the boss and the third accomplice ("The Eel", who doesn't do much during the story) come out of the house, Tango blows his whistle "loud and long enough to bring all the police in Paris" and yells at them that they're under arrest.