You have two different frame materials between the Henry and older conventional leverguns.
The Big Boys use a brass/bronze alloy that's stronger than just brass, while the Marlins & Winchesters & so on use steel.
Frame stretching over time with hotter loads in any caliber will be more likely to happen in the Henry than with a steel frame using the same loads.
How much & how soon you'd experience any frame stretching is unknown.
I liked the Henry Big Boys I tried out, they were smooth & accurate, but I would not want to carry one in the field and I would not want to run a high volume of hot loads through one.
You might email Anthony Imperato through the Henry website & ask his opinion on the Buff .44 Mags. He's pretty forthright.
Depending on where you live, 300 yards is a long shot for hunting. Dense forest will be much closer to 75-150 yards, open desert or high mountains will extend that.
Neither the .44 Mag or the hot .45 Colt is a good candidate for much beyond 150 yards.
If you want a .308 levergun, your choice is a Browning.
What do you want your lever-action to do?
Handgun calibers can be extended range-wise in a rifle, but they're still handgun cartridges.
You want distance & power, you'll have to look into a centerfire rifle caliber.
Marlin makes a couple leverguns that use new loads to achieve greater practical working distances, the Henry Big Boy wasn't built for that.