I have rifles in both the 338 Federal and 358 Win (both in levers and bolts). I love them both, having gotten my first 358 Win (a pre-81 BLR) in 2003, and my first 338 Federal (a LH Tikka T3) in 2015.
I built both of my 338 Federals as I am a LH shooter. The lever action is a rebarreled Winchester Model 88 with a heavy 20" Krieger barrel, and the other is a rebarreled Sako Model 85 Finnlight II. Rocky Mountain Rifles (Corlanes) did the work on both rifles for me.
The Model 88 was built for brush busting and is set up with QR rings and bases, and I have a peep that goes on its rail, paired with a NECG banded front sight. There were some feeding issues with it at first, with the action and magazine (originally a 308 Win) but were resolved. I use Federal's 210 gr Nosler Partition ammo in this rifle and it will produce 0.893" groups at 200 yards. It gets 2505fps out of the 20" barrel, and retains 1935 fps and 1746 ft lbs of energy at 300 yards, dropping 1.3" from a 200 yard zero. This will provide good expansion and penetration on bears, moose and elk, yet with 2116 fps and 2087 ft lbs of energy at 200 yards is still legal for bison. It has taken a very nice caribou to date at 158 yards.
The Model 85 was built for typical hunting varied terrain, with a 22" Wilson carbon fibre barrel (1:10 twist). The original throat was an issue, but we had it rechambered with a reamer with a longer throat, that cleaned up the accuracy issue. At the moment, I have been using various factory ammo in it, and the 200 gr Trophy Copper load is the most accurate for me with 0.457" groups at 100 yards. It gets 2579 fps out of the 22" barrel, and retains 1865 fps and 1544 ft lbs of energy at 300 yards, where it drops 27.4" from a 200 yard zero. It retains 2205 fps and 2160 ft lbs of energy at 200 yards for bison. The 200 gr Uni Cor ammo produced 0.477" groups for Corlanes at 200 yards when they fixed the throat issue. I can produce 0.654"" groups with it at 100 yards. It gets 2687 fps, and retains 1931 fps and 1655 ft lbs of energy at 400 yards, dropping 25.3" from a 200 yard zero. While it doesn't quite carry the energy required for bison to 300 yards, it is still legal at 292 yards. It has taken a couple of bull moose, a caribou and a stones sheep to date, at distances ranging from 25 yards to 150. It also shoots the Barnes 185 gr TSX ammo into 0.821" groups at 100 yards at 2700 fps. It will retain 1830 fps and 1376 ft lbs of energy to 400 yards, dropping 26.6" from a 200 yard zero. It is still legal for bison at 200 yards with 2241 fps and 2063 ft lbs of energy. I plan to develop a load for this rifle with the 200 gr AccuBond.
As mentioned above, my lever action in 358 Win is a pre-81 BLR, that has been carried far more than shot at game, as it is my main camp rifle, back up rifle for guiding, hunting, archery hunting, and for carrying on a horse or ATV. The trigger is terrible, but I know exactly where it breaks in its creep, and can consistently produce 1" groups at 100 yards with its favourite handload with the Speer 220 gr Hot Cor FN at 2201 fps out of its 20" barrel. It retainis 1865 fps and 1699 ft lbs of energy at 200 yards. It isn't legal for bison, even at 100 yards, due to its slow initial velocity. I recently tried the Hornady 200 gr Interlock ammo in it, and it produced its smallest group to date of 0.510" at 100 yards! At 2469 fps, it will retain 1933 fps and 1659 ft lbs of energy at 300 yards, dropping 3.3" from its 200 yard zero. At 100 yards, it is at 2194 fps and has 2135 ft lbs of energy making it legal for bison. It has accounted for mule deer, moose, elk and a grizzly bear over the years, at distances ranging from 20 to 140 yards.
I now have a LH Rem 700 in 358 Win! Initially, it had a REM DM, but the magazine was not designed for the 358 Win cartridge and would not feed or retain them properly. This was switched out for the regular BDL magazine box and floorplate. So far its 22" McGowan barrel (1:12 twist) prefers the HSM 225 gr SGK ammo, producing 100 yard groups of 0.557" at 2405 fps. This retains 1924 fps and 1849 ft lbs of energy at 250 yards, dropping 8.5" from a 150 yard zero. This would be legal for bison at 200 yards with 2015 fps and 2029 ft lbs of energy. I am still working to develop a handload with the same bullet. Both W748 and CFE223 are showing promise, but have yet to try my test loads with TAC and IMR4320.
While I prefer to have loads that will carry the velocity and energy to longer distances, should it be required for any circumstance I am likely to encounter my targeted species at, over the past 38 years of hunting, my average shot distance has been just 132 yards, so I really don't need the additional ballistic capacity of the 338 Federal over the 358 Win. I just have more varied experience on game with the 338 (338 Federal, 338-06 and 338 Win Mag) vs the 358 (358 Win and Whelen) and therefore a slight preference for the Federal's capacity. Love the 358 Win, and will continue to use it, as I will with both cartridges for years to come!
Whichever the OP chooses, I wish him luck and many great adventures with a great cartridge that outperforms its papr ballistics in the feld.
You can look at the NECG sights for options where you won't need to dovetail the sights into the barrel, mitigating barrel profile concerns.

