Not directly applicable, but related:
During playtesting, I asked Vincent if the Driver's car bonuses applied to the optional battle moves. It doesn't say they do, but the logic seems to imply that the moves gets you bonuses to all relevant basic moves, of which the optional battle moves are a part (as in, you don't have to be no one special to do them). Vincent said yes, the bonuses apply.
If you look at it like that, the question is whether the custom move is or is not basic.
Custom moves are always more specific than the original basic moves, but they can be in the same-ish area, as you noticed - "when you pickpocket something, roll +appropriate stat" is going to be different from "when trying to catch Ratsie in the warrens, roll +cool" applicability wise.
So say you have a kinda-sorta basic move, but it has that tinge of specificity that that doesn't allow you to just let someone use a stat without anything - and when you do, you can bridge that gap. Ask the Battlehardened gunlugger if he thinks his move would apply to refined act under fire pickpocketing move. If he says no, cool. If he says yes, ask him what kind of battles teach a man to steal. Listen to answer, reincorporate later if you want. Usual MC fare.
You might end up with a situation where Battlehardened does different things from game to game, but I'm not sure if that's that big a deal.