With the quality muzzle brakes
we have today, recoil is not even a consideration. I have had 13 year old girls shooting my 338 and 375 Allen Magnums and slapping my 20" diameter gong at 1000 yards and beyond so as far as recoil, not an issue.
For those that say if you have to have a muzzle brake
, your shooting to much gun, will, lets just leave that one alone for now, simply put, silly opinion to have pure and simple.
For what you list, moose, black bear and whitetails, you really do not need a 338 magnum chambering. Nothing at all wrong with them but moose hurt very easily. In all honesty, moose are generally shot at pretty close range and they are no where near as tough as elk are.
If your looking to hunt elk past 1000 yards, certainly get a 338 magnum chambering as a first choice option but for all else, its not needed unless you want one. I do not believe you can be overgunned but you do not ALWAYS need alot of horse power.
That said, I DO believe you can be UNDERGUNNED. The 243 is a great chambering but in my opinion, for deer hunting, you would be best to keep shots under 400 yards. I will take some heat for this comment from some but it just does not have the bullet weight or retained velocity or energy in my opinion for hunting much past this range on deer size game. Will it kill deer at longer ranges, certainly, MUCH longer ranges but I would not do this on a regular basis.
IF a customers came to me with the requirements for a custom rifle
that you want. Probably my first choice would be the 300 RUM loaded with the 200 gr Accubond at 3200 fps in a medium weight rifle, 9-10 lb sporter with 26-27" barrel length. More then enough power for anything you list, very accurate, easy to load for, no fireforming so you can just get started focusing on making good ammo and then shoot.
Recoil, a 300 RUM sporter rifle with my PK muzzle brake will recoil less then a 243 sporter weight rifle, more muzzle blast for sure but a good set of electronic hearing protection will solve that problem and they work great for hunting situations, especially long range.
The 300 RUM offers a huge option list for bullet choices, relatively good on barrel life for a big game rifle, huge selection of powders to use, decent brass at decent prices, very accurate, very powerful and great terminal performance on the game you list.
That would likely be my first recommendation.