These caliber comparisons often evoke opinions and argument but I have a few thoughts based on experience with the .358. There is some consensus on the .358 on game, you might consider it.
I was fortunate to buy a Ruger in .358 back when nobody chambered for it, saved me the cost of a barrel or rebore. I absolutely thought then, as I do now, that this is a superior woods cartridge out to 200+ yards and maybe further.
I have shot a lot of deer with this cartridge as I have hunted several states for many years, and I will say something others may argue with: I think there is something special (killing-wise) about a mid bore running 2200-2600 fps. The one-shot knock down kills I have accumulated are amazing. I grew up witjh a .35 Rem, a round with substantial thump inside 100 yards, I shot a lot of bucks with it. You guys that hunt with them know that it works. The .358 Win. is that on steroids without being a magnum and all the stuff that magnum means.
I can't speak to the .338 Federal, I understand the sectional density of the bullet is superior, but as a practical reference I have only recovered one bullet from a buck shot lengthwise out of dozens with the .358. Not sure if the SD differential is relevant at normal range.?
Me, today, I would go with the .358. Readily available cases, powders are common, really easy round to load for and not a fussy cartridge. Mine shoots a variety of cast and jacketed, just a democratic kind of cartridge... pretty much likes every load but will let you know when you hit the sweet spot.