The bristles have a locking mechanism between it and the butt of the rifle. So it’s detachable.
I assume firing while flying is a ‘last resort’ type of combat, and not the intended use case for this kit. If the rifle wasn’t chibified, it could be made long enough the stability of the butt would be done by the users thighs, right hand on the trigger, and left hand on the barrel/grip. You’d need a secondary sight at the end of the barrel to account for this pose, or some sort of magical version of it.
Looks like this is less of a skirmishing frontline kit and more of a mobile stealth / sniper setup, where the witch goes around finding a good spot and getting set up to fight before rapidly leaving. Take the hat off for reloading, then quickly reattach the bristles and put on the hat before breaking contact and flying away.
I do like your skirmisher motorcycle analogy. That does seem the most likely.
There is also a bullet magic grimoir, which is explicitely included in the kit. The character very likely has some sort of aim/damage/range bonuses that get applied during combat, if not outright magic missile levels of targeting.
If the kit is for skirmishing though while staying mobile; yeah. Bullet hat is dumb.