All of it. Let them have all of what they lay hands on.
There is one fundamental rule for character advancement in Apocalypse World: change is both prescriptive and descriptive. It's the very first rule in the Improvement chapter, page 178. What does that mean though?
- Prescriptive: When something is added to the character sheet, it becomes true in the fiction.
- Descriptive: When something becomes true in the fiction, it is added to the character sheet.
This means that when the Savvyhead repairs an LMG and cobbles together battledress, those things go on the Savvyhead's sheet and now the Savvyhead has an LMG and battledress.
That would explain why it's your main dilemma: when you resist this, you're actively working against a basic feature of the game's design.
Having stuff is fun but does not make one king
Besides, having an LMG and battledress does not in any way make a Savvyhead into a Gunlugger. Nobody can do what a Gunlugger can do with that stuff — the Gunlugger is still the Gunlugger and outclasses everyone at being a one-person army.
Becoming someone else is always an option, too
Yeah, so there might already be a Gunlugger, and now the Savvyhead is gearing up, choosing get a move from another playbook and taking Gunlugger moves to be more bad-ass, and otherwise encroaching on the Gunlugger's protected “niche”! Oh no, what now?
Nothing! You do nothing. Who and what the PCs are is their job to push for, and if the Savvyhead wants to try to be the Gunlugger, they can do that. It's not free though — they get what they manage to grab via their efforts in the fiction and the moves they take when they Advance, not a perfect cloning of the other Gunlugger just by their say-so. They're going to be different because of how they got there, so whatever problem is being anticipated here is imaginary, not real. The Savvyhead could even swap playbooks with The Ungiven Future (p. 182) and it would still be peachy — not only are they going to do “Gunlugger” differently than the existing Gunlugger, but who says there isn't room for two big, gun-toting bad-news girls in the game? There's no reason not to — they just get to share some of the same kind of fun, and you can have bigger exploding firefights with them.
Easy come, easy go
Gear, however, is vulnerable. You have the MC move Take away their stuff (p. 116). Use it! Don't use it gratuitously of course — set up hard moves with soft moves, or wait for a Golden Opportunity — because remember that having stuff is fun and they earned it, somehow. And remember that whenever you make a move you must obey your Principles (p. 110), which means that even when you Take away their stuff you still have to Be a fan of the players' characters. So when you take away their stuff, do it because you love them and that's the most honest thing for spurring them toward awesome gameplay that moment, not just because you don't want them to have that stuff.
So: if they can work to having it, they can work to losing it just the same. This goes for even stuff that comes with the playbook: a Gunlugger can drop their big gun off the Bridge Between Holds, a Driver can lose their beloved hot-pink-painted heavy-assault school bus when the bomb someone planted in it goes off. (Of course, they're going to want it back/fixed/replaced, and how they do that is a beautiful driver of more gameplay. See pages 152–8, “Moves Snowball”.) What happens to gear in the fiction goes right back to the rule that character sheets are both prescriptive and descriptive — when something gets toasted, lost, or stolen in the fiction, you change the sheet to reflect that, and that absolutely includes the precious gear they start with.
They'd better not bounce around in the world thinking that this thing has Plot Protection and that thing doesn't, should they? That's the point: nothing has plot armour, and the world should feel real, and the players have to avoid believing that the fiction is going to bend around the mechanics to maintain their special snowflake gear — keeping their gear safe and in good working order is their job, not the MC's job.
Remind them (p. 96):
Your job is to play your characters as though they were real people, in whatever circumstances they find themselves in — cool, competent, dangerous people, but real.
That includes not indulging in magical thinking about their stuff or character sheets — nothing has plot immunity, not even the PCs, never mind their mere stuff. Again, this goes right back to the fundamental rules of play, descriptive & prescriptive, page 178. The examples there are worth re-reading: this can involve potentially heavy losses, but that's real.
There is no status quo, there is no niche protection
If you're used to PCs having guaranteed niche protection, you're going to have to sit down and rethink some assumptions about Apocalypse World, maybe do another read-through of the book. Just because the playbooks look like classes, they aren't and don't work like them, and they provide zero niche protection once the game starts.
Just like the game says, there is no status quo. The players are given very powerful options for changing the world, and that includes themselves, each other, and how they interrelate. And if that means that Doom the Gunlugger gets her nose out of joint because Lemieux the Savvyhead went all Iron Man and then swapped to the Gunlugger playbook? Well then, they just handed you 24-karat MCing gold, didn't they? Poke that rivalry and see what happens!