Not sure if you've worked out what to do with this action yet, but here's a wildcat idea for you:
[myself, I'd do it just like this, but in .358 caliber.]
348 Winchester (base dia about .55"), trimmed to 2", necked to .375, shoulder at about 1.475" and .533" diameter, 30 degrees. .355" neck.
With seated bullets, there's about exactly the same capacity as the 350 Rem Mag, only you can load this puppy much hotter (52k CUP instead of 40k or so?)
The 350 gets 2350 fps with a 250gr bullet, so I'd bet you could get about 2500 fps or more with a 270gr .375 bullet (such as Barnes Original FP), and up to 2800 fps in a 225 gr (Hornady Interlock FP).
In other words, more than you'd want in a 7 1/2# rifle!
Here's some ballistics for you:
The 225gr, with an approximate BC of .235 leaves the muzzle at 2800 fps and 3917 fpe. Sighted in at 200 yds,
125yds = +2.3", 2755 fpe
250yds = -3.9", 1880 fpe
300yds = -10.1", 1600fpe
The 270gr, appr BC of .260 going out at 2500 fps and 3816 fpe, 0@200,
125yds = +2.89", 2735 fpe
250yds = -5", 1885 fpe
300yds = -12.9", 1620 fpe
That would be the meanest Big Bore around, I'm sure! Definitely turn some heads. A damn fine elk gun if that's your gig.
-oh, and be sure to put in a good muzzle break!