I built a 9 lb .338 Norma mag, and run 270's out of it now, with a 24" barrel. Loading very conservatively, I am getting 2830 with 270's and 2630 with 300 Bergers, I could push it more and still be well within the pressure limits of the brass, but I am keeping my loads low/moderate as it's on a standard remington action. With a .750 bolt I could run up to the full capabilities of the brass and likely be 2750 or so with the 300's and pushing on 3000 with the 270's.
My rifle with a can would be about 10-10.5 lbs, but to be comfortable to shoot, I would likely look at the 230-250 range of bullets. I have only ran it with a good brake, so I can't say for sure, maybe the 270's would be alright, or the 265 ABLR's. But with 88-92 grains of powder, that is a good bit of horsepower in a pretty light rifle. Could be doable, but would be on the upper level of recoil.
However, a .338 Sherman Short or Sherman Max may be the perfect solution. They run the 250's around 2850 fps or so on the upper end in a 26" tube, with 60ish grains of powder, and run in a short action. The Sherman Max would be 2900 or better on the upper end, but would require a medium length action, or possibly get away with a Bainey 3.110" box. These would react very well being supressed I bet, as they are pretty efficient and use 20-30 grains less powder than the rum's and Norma mags, so would likely be pleasant to shoot. You would probably hit 2700+ with 250's and a 22" tube. The caviat being, you must be willing to fireform. The max version makes it simple, it is the same process as an Ackley Improved fireform. There are Hornady dies available for $180 range, and quality ADG headstamped brass.
For a short barreled lightish rig to run supressed in .338 for mountain hunting, that is the route I would go.