The 338-win mag & 338-Remington Ultra Mag are the two rounds you should look at the hardest.
One more 338-caliber round to consider is the 340-Weatherby. I own & hunt with a Sako rifle that is chambered in 340-Weatherby, I love this gun.
The 338-Lapua is a awesome round, but you will have a heck of of a time finding one light enough that you want to hunt with it.Sako used to chamber the 338-Lapua in asporter/hunting style rifle. I have never seen factory ammo for the 338-Lapua.

The 338-win mag will have the cheapest factory ammo. The 338-win mag riflemight be a few dollars less than the 338-Remington Ultra Mag & 340-Weatherby?
The 338-Lapua???
The 338-Federal, I am not sure why they even developed this round? I would steer clear of it because it might die off in a few years? Then ammo or brasswill be hard to find.