For medium ranges on larger North American game, effective cartridges are normally looked at STARTING with the 7MM Rem or .300 Win Mag and go up from there. For moose & brown bear, most would say start with a .338 Win Mag, .340 Weatherby, or larger. Some of the modern mags, like the .28 Nosler, pack a lot of punch into smaller calibers, so that may be worth considering if you take the large brown bears off your list.
As for recoil, every cartridge worth considering could loosen your fillings in one rifle and be comfortable to shoot in other rifles. FWI: I picked up a used Weatherby AccuMark in .338/.378 Weatherby Magnum that is as comfortable to shoot as a .308, but it's not a light-weight, and it has a brake. The only limitation on shooting that rifle is the CO$T of ammunition. (It's devastating on elk!) The Weatherby Alaskin in .300 Weatherby I picked up is as comfortable to shoot as a 6.5 Creedmore. It doesn't have a brake, but it's not a lightweight rifle.
Next thing to consider is cost and availability of ammunition. If you are looking to use factory ammunition, you need to limit yourself to cartridges you can readily buy ammunition for. You aren't going to find PRC or Nosler ammunition in many stores. .300 Win Mag ammunition is normally available almost everywhere. Weatherby ammunition is normally available in larger sporting good stores. It used to be we could ignore ammunition availability, but not for the last 6 years.