I'm not sure putting snow down someone's back rises to the level of assault. Sure, there are huge class issues there in what an elector is permitted that a worker might not be and even what a worker can do that an unemployed man is unlikely to get away with, but let's be honest - Bambi called her names and put snow down her back. Fantine drew blood. Fairness would probably prescribe not pressing charges on either one, but if charges are to be pressed, Bambi was a dumbass and a jerk, but Fantine committed immediately verifiable harm.
Bambi caused more harm, maybe, but I can't think of a functional legal system that would punish not even very clever insults and a fistful of snow down a person's back more harshly than jumping a guy and digging your fingernails into his face. Yes, she was retaliating to provocation, and six months is ridiculously harsh, but the most either of them ought to deserve is a night in the drunk tank, maybe an additional writeup/warning/couple of days for Fantine to cool down. Javert only saw the fight, in any case, and Fantine has plenty of motive to exaggerate the provocation; I do wonder if in Javert's head, he's seeing something more like the situation in the musical, an out of hand argument between whore and customer, because he's got to be thinking "You did this to an elector because he put snow down your back? Either you're a crazy bitch who needs locked up for the protection of the city, or you're a liar who won't admit to flawed business practices and need to be locked up for the protection of the city."
What kind of literature and what kind of life is the same question. - Tom Stoppard