A quick look in a reloading book show me the .444 is .453" at the case mouth, and .470" at the case head. It looks like a straight case, but it isn't.
The .44 Mag is shown as .456" at the case mouth and .457" at the case head.
SO, while you could likely neck size .444 brass in a .44 mag die, the specs say the .44 die is .003" too large AND .013" too SMALL for the .444 case head, assuming you could get it all the way into the .44 mag sizer die.
I'm not saying you couldn't make ammo that would fire, but if you want the best, then you need the right tools (dies) to make it.
Don't try to cheap out, dies are not that expensive. GET (and use) the CORRECT reloading dies for the cartridge.
All else being equal (and it almost never is) bigger bullets tend to work better.