Owned a marlin 1894 for years with the same 1:38 rot and it was fine for bullets between 180 - 265 grains.
One of my most accurate loads was with the 265 grain hornady fp and h110/w296 powder even though the bullet was designed for the 444 marlin.
The 240 grain factory ammo is fine for deer at the relatively short ranges the firearm was designed for.
Just bought a henry big boy all-weather in 44 mag myself.
Stick with bullets between 180-265 grains and you'll be fine with that rifle.