Will a shortened 444 case fit in the chamber of a 44 mag?
Sadly, no, the .444 is .4698 at base, .44 is .4569.
As stated above you can use .303 British or .30-40 but you have to reduce the rim.
.445 Super Mag is easiest, I ran them into a 7x57 die to reduce the upper part of the case enough to chamber. I think it would work better if I had a .41 Magnum sizer, but after firing they fire form to look exactly like JJHack's in the OP. I loaded mine with a .410 shot cup trimmed to length and silicone caulk to seal.
Oh, yeah, .445 works in both .44 Mag and .44 Special if you size down the "neck" far enough.