For stretching shooting distance the single most important item, besides the cartridge and rifle, is being able to shot consistently. The greatest innovation for increased consistency, for me, is the muzzle brake
Pushing 300s as hard as I could in a Ruger #1 was only comfortable for off hand shooting. I had no nifty stuff hanging on it. Even had that thin red Ruger butt pad.
Right off the bat I'd suggest a brake such as the Muscle Brake by Jim See on this forum. With Kirby's PK brake on my 375 AM results in not much more than a twitch when she ignites.
Your barrel is of a good length for stretching things a bit.
The key to stretching it is "bc" and velocity. A balance between bc and velocity may well be key to success.
Do some computer runs with JBM Ballistics or some such program to see what turns out best.
Here are some bullet selection suggestions:
Hornady 300 gr - 0.460 G1 bc
Nosler 300 gr - 0.485 G1 bc
Rocky Mtn 370 gr - 0.926 G1 bc - Most probably too heavy to get adequate MV.
Sierra Game King - 300 gr - 0.475 G1 bc
Speer #2472 - 270 gr - 0.429 G1 bc
I'm thinkin' 1800 fps would be about max range to ensure expansion. But with the amount of FPE remaining I think I'd go for bone with the large 375 bullets.
Keep us posted