What long range, accurate cartridge for elk from five choices. See below.
Has - .308 Win, Most powerful rifle currently owned.
Possibles - .280 AI - Of the ones you list this would be my pick.
.28 Nosler - I know nada about this.
.300 Win Mag - I should like this as it has a lot of development behuind it but I don't care for them. My .308 Norma and .300 Weatherby bracket it well. I use a .338 in this role.
.340 Weatherby - For a boomer, I would prefer this over the others listed and the Lapua simply because I can chamber it in a Model 70 action. Try to get a minimum chamber but with the freebore if you go for one .
.338 Lapua - Love this round but the Sako I have access to only has a 26 in barrel it should likely have at least a 26 in. .
Options to minimize recoil like a brake and barrel life are a concern. I wouuld forget the brake. I put one on my .338. It kicked kike a .270 but the blast was ridiculous. The brake is now in with the loading dies for the cartridge. It will stay there.
Lots of folks don't want to be around anyone using a brake. I am in that group. high quality recoil pads help a lot. So does rifle weight.

From my preferences? A big .30, likely pushing a 210 gr ABLR at 3000 fps or so. I am going to see how my Weatherby likes them. Starting new? likely a .300 PRC.