I've been thinking about your idea of a 44 Mag (0.431”) bullet in a standard (0.473”) rimless case (308 Win/8mm Mauser/30-06) and think you had a pretty good idea.
You could pick up an un-chambered Douglas barrel, already threaded for a Mauser Large ring action, and already with a #5 Sporter barrel taper for $250, use off-the-shelf 444 Marlin chamber ream and dies, and use 30-06 brass for your fire-forming. If you used the whole 63mm -06 length case, the ballistics should be pretty close to a modern loaded 45-70 at 3000ft#. Not a bad idea.
Nossler Partition 300gr bullets, Swift A-Frame 300gr, Sierra Sports Master 300gr all look like excellent bullets for that cartridge, and would have equivalent sectional density to a 150gr 308 bullet. You would get a lot of bullet arc, so this would be a 250 yd or less rifle, but inside 250 yds, you would get plenty of knock-down for most any critter.
So I did some more looking and found a 10.75mmx57mm Mauser-Mannlicher designed for big game in Africa. 350 gr bullet, 1950ft/sec, 2960 ft# ME developed around 1900, very very rare. People have tried this before, this cartridge has no shoulder, so would have to head-space off the cartridge lip like a 9mm or 45 auto pistol cartridge, but obviously that is not insurmountable.
I'd be very interested in how this works for you, if you try it.