Time to turn the Nashers against the Dead Rats. Easy. Like most insulated groups, they're paranoid that their enemies control neighboring factions. In this case, they'll be correct.
There's a rabble-rouser preaching from a fountain farther into the district. He's an effective recruiter, popular within the Nashers. If he were killed, the Nashers would be outraged. Seeing how I want him killed, the trick is directing that rage at our enemies.
Kill the rabble-rouser, and plant the Dead Rats dagger on the fountain. That should get both factions at each other's throats, allowing my agents to establish some order in the aftermath.