This is most likely crime lingo, referring to a cocaine addiction.
Cocaine is often snorted and sometimes referred to as 'nose candy'. What he means is that they're basically exploiting someone's drug addiction (whether by supplying him the drugs or blackmailing him about it) to get him to help them. This is a fairly common trope with both criminals and cops (who would sometimes use a drug possession charge to flip witnesses).
To further corroborate this, notice that Buddy says he and Darling will be back for another job once their 'nose bag' is empty - and Bats later explains exactly what that means:
I know you guys got noses to feed... You rob to support a drug habit.
[And as for this has something to do with Eddie No-Nose, don't ask me that!! ;)]