Whichever way you go, just make an informed choice considering the distance you'd need the gun to do & the amount of usage it'd get.
Trust me- If you do ground hunting on foot, the Big Boy's weight WILL be an issue at some point.
The ones I shot were nicely done, but I couldn't see myself carrying one for very far.
200 yards is unrealistic for a Henry in any handgun caliber.
The .45 Colt in standard pressure loads should run through a Henry for many years & they're fairly cheap & easy to handload (inexpensive lead works just fine). Hotter .45s run into the same potential for frame stretching as the .44 Mags, and they're more expensive to buy or handload (jacketed bullets, usually).
The Browning (I've tried one in .308) is a well-made levergun (any prejudice against it being made in Japan is idiotic, that Miroku factory turns out better quality guns than most of our US companies do), and it's easily capable of 200-300-yard shots with adequate power and accuracy.
The frame there is aluminum nowdays, but the design is a thoroughly modern gear-driven system that locks the bolt into the rear of the barrel, and the frame doesn't take much stress (One Browning rep told me the frame on those guns "could be made out of cardboard" when we were initially setting up a loaner & talking about alloy vs steel.)
I'd be interested to hear what Anthony has to say, myself. We talked about the Big Boy when it was first introduced & a couple times since. I have no worries about the strength of the frame's bronze alloy in what I'd consider normal usage, but high pressures combined with high volume can create the risk mentioned above.
Just make sure you either email him or ask for him direct on the phone, I'm not sure what you might get from anybody else there.
There are better hunting rifles for 100+ yard terrain than the Big Boy (or any conventional handgun-calibered levergun), but in the forest if you can learn to shoot well with iron sights you can be effective with one inside that distance & slightly beyond it.
Generations of hunters used the Winchester Model 92 in .44-40 as a deer rifle & brought home meat every time they went out.
On leaving the Big Boy loaded, the only risk there is a possible (I said POSSIBLE) set developing if the mag spring is left compressed over long periods of time.
If that happens, it COULD alter feeding reliability.
Leaving the mag one round short could help.