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.44 magnum is suggested by Referee's manual to have damage 2, crit 2, armor +1. This kind of makes sense when you think about in the sense of big, powerful revolvers and doesn't really unbalance things that bad because revolvers have terrible range and mag capacity.

But there is of course rifles chambered in .44 magnum, notably reproductions of wild west style lever actions. A rifle with a decent range and 10 or so rounds in the magazine would be incredibly lethal, far outperforming 5.56 and 7.62 (at least against unarmored targets), which seems slightly absurd.

Dam 2, Crit 3, Armor +1 seems to be good place to put it, with worse range than 5.56.
Of course that's where .45 ACP is at in the core rules, and reasonably where you'd put .357 magnum (a far more common caliber).

Currently I'm considering dropping .45 ACP to 2/3/+2 and putting .357 at 2/3/+1 but I have no idea where to put reasonably put .44 magnum.

I wouldn't change any of the extant cartridges and I agree that, at this system's level of granularity, .357 Magnum having the same stats as .45 ACP seems right. As you note, .44 Magnum already has stats in the Ref Guide.

So what would a .44 Mag Henry rifle look like?

There is debate about the effective range of a .44 Magnum rifle, but we'll bias to the lower end, in line with other design decisions in the game. So, assuming 150m, that gives us a Range stat of about 2. Being a rifle means it's at -2 to hit at bad breath range and readily usable as a club. Using the lever-action as a base, we get

Rifle/.44 Magnum/ROF 1/Dam 2/Crit 2/Blast --/Range 2/Mag 10/Armor +1/Weight 1/Price 225-ish

It's not super and it needs at least 3 successes to deal a Crit through body armor, the same as an M4.

I wouldn't change any of the extant cartridges and I agree that, at this system's level of granularity, .357 Magnum having the same stats as .45 ACP seems right. As you note, .44 Magnum already has stats in the Ref Guide.

So what would a .44 Mag Henry rifle look like?

There is debate about the effective range of a .44 Magnum rifle, but we'll bias to the lower end, in line with other design decisions in the game. So, assuming 150m, that gives us a Range stat of about 2. Being a rifle means it's at -2 to hit at bad breath range and readily usable as a club. Using the lever-action as a base, we get

Rifle/.44 Magnum/ROF 1/Dam 2/Crit 2/Blast --/Range 2/Mag 10/Armor +1/Weight 1/Price 225-ish

It's not super and it needs at least 3 successes to deal a Crit through body armor, the same as an M4.

This is reasonable. I'd put it a range 3, to give it an edge over pistols (where range 2 on the other hand is quite generous). It still irks me that the "automatic crit" takes down a bear with ease, but with the granularity of the system there's not a lot of options. Besides, that can be resolved by tweaking the hunting rules, like giving bears "armor" to simulate their resilience or adjusting how crits work on animals. If it's even worth the bother.