Really, though, I can’t imagine that most portable anti-tank weapons wouldn’t be suitable for the task. Even just the RPG-7, with it’s variety of warheads.
I would recommend shooting low with a large caliber machine gun to limit mobility of the puppets, then applying the napalm.
If you have any LOX available, and you have a reliable delivery system (I’m sure Frau Totenkinder might be able to whip something up), that may be a good follow up to the napalm.
The problem with axes, is that you have to get up close and personal. The puppets are much stronger than you, and there are more of them. You might get a couple of whacks on one, but that’s about it.
If it wasn’t for the low rate of fire, old-fashioned 6-pound cannon shot would be ideal: a big heavy projectile with enough momentum to smash anything wooden to flinders. Hmm… in modern terms what you’d need is something like a 75-125mm autoloading naval gun, only with shortened barrel length and solid rounds with reduced muzzle velocity.
I heartily approve of this use of the word “flinders”. It’s a fine, fine word.
Presumably you could harpoon the wooden warriors in some fashion and then drag or hang them so as to remove them from being a threat, but that would require more than a handheld weapon.
Personally, I’d start by looking at ways to impair them rather than kill them outright. Even a human made of flesh is going to be pretty hard to kill if all of his pieces can keep going when they’ve been chopped off.
Since you can stab things into wood pretty effectively, I’d play around with grappling hooks, pikes, spears and lances. A wooden enemy won’t be all that effective if he’s stuck on the end of a pike, Or if he’s entangled with a grappling hook stuck in his back. Some of the effectiveness here may depend on the size and strength of the wooden soldiers, though.
Once they’re at least partly immobilized, go to town with axes, guns, fire, etc. or just run around them if it’s not necessary to kill them.
I do think the OP underestimates the ability of guns to rip up wood. The biggest gun I’ve ever fired was only a buffalo gun (.45 caliber), but we took a 2" hole out of a sheet of thick plywood with a single shot. Make that a heavier machine gun or shotgun and you’re set.
Chain shot would just stick into them, giving them sharp metal studs all over. That would help if you could get them to fall down on the carpet, at the expense of turning them into walking rasps.