The 44-40 is likely the best choice for off the shelf ammo,,if there is a 'good' choice for doing this.
Another thing to explore and it'll tell you very quickly if it's do-able or not..
Many 444Marlin chambers will accept a 45Colt (Long Colt) case.
Some will not as the 45Colt base is ever so slightly larger than the 44-40.
Worth a try I think.
If it does, you can neck down the 45Colt case to 44cal. Seat and crimp a .429 bullet and fire it in the 444Marlin chamber.
You will be doing pretty much the same as what the 44-40 rd is doing for you.
You will end up with a straight blown out 45Colt case for your efforts.
45Colt brass is generally cheaper that precious 44-40 brass.
Many have 45C brass around, at least some of it.
Call it the 44-45 Marlin & Colt.