I started thinking on this by contemplating a subsonic .338 with the 300 grain bullet. Everything looks just fine, but it was pointed out to me that subsonic cartridges are very hard to impossible to tune.
Keeping above the transonic region at 200 yards means yes, quite a bit stiffer than a .30 BR. Believe the recoil would be about 7 foot pounds in a 13.5 pound rifle. We're talking about something more or less equivalent to a .35 Remington, or a .308 with a 180 grain bullet at 2,600 fps, when fired in a 13.5 pound rifle -- not a sporter!
Now the barrel dwell time *might* be an issue, but it should be tunable.
One real issue would be torque. You could cut the twist needed considerably with a .250 Hybrid -- so less torque -- but the drift gets pretty close to the .30 BR. You would save about .5 inches at 200 yards in a 10 mph wind, but who shoots that if they can avoid it? Bullet still makes a bigger hole; not sure it's worth it. Recoil would be around 5 ft/lbs in a 13.5 pound rifle, which is not too much more than a PPC in a 10.5 pound rifle.
The looking good on paper part -- I did say I've shot this bullet at 100 yards. It was a 17-pound rifle with a brake, but the load was also equivalent to a .338 Lapua magnum -- 2750 fps. There are things more fun in life, but if you know how to shoot one, it's not too hard with rests, off a bench. I wouldn't want to shoot it prone or sitting.
Probably won't do this one, just too many other projects going on. Sounds like no one else is a believer.
Edit:
I did talk to Randy about this. I don't think he felt it had much promise, but he heard "That'll never work" too many times with the .30 BR to be dismissive. He believed it would be quite some time before Berger offered .338 jackets. I wouldn't expect him to make the bullets, either.
But the 300 Hybrids measure, and shoot, very well indeed. Fully the equivalent of a custom, given their size.
Edit No. 2:
Just to show prototype possibilities, I *think* a .338 Wolf Pup would do it. I don't have QuickLoad.
You'd need a special bushing to use an existing .30 Wolf Pup reamer in a .338 barrel, then take the neck out with a with boring bar, or even a twist drill. Yeah, I know, crude. Been done before. Skip the FL die at first -- we're not talking terribly high pressures -- and make a seater & neck die. Total tooling costs -- the bushing that fits the .30 Wolf Pup in a .338 barrel.