Re: Which .338 should I have built?
Was away for a bit so not ignoring your question. I noticed that you have decided on a big case 338. I like the RUM based one and have my own wildcat called the 338 Mystic.
Back to the question if a heavy 7mm bullet will do the job at 1000yds. yes, it most certainly will. The key is impact velocity vs bullet toughness. When that is in balance and the bullet arrives in the right place, animals die.
Am I infering that the 162gr 7mm is equal in hitting powder as a 300gr MK 338? Not in your life. That is big mojo. Do you need it? Up to you and your budget.
Just remember that you must have a barrel and case large enough to get over 2800fps with that 300gr slug. I am not enamored with the ballistics of lighter slugs. You still need impact velocity to make that bullet work. Failure to expand means you now have a 338 FMJ. Not much hitting power there (at this point the 7mm would work better!!!). Either of your choices WILL WORK in a long enough barrel.
now the tough part. Getting the right brake. Do not use a Vais style. Reason? You want the most effective brake you can install. With a "light" 338 cannon, you are going to have to deal with massive recoil. Even more important, you need to spot your shots. That brake better reduce recoil AND keep that muzzle down and/or going straight back.
You will be double plugged (you should be) so noise is not an issue.
Best designs are very wide fishgill type brake with no forward venting. Example would be the Micaluk/Holland or that ugly wonder on the AR50. I would suggest a brake at least 1.5" wide (2" better, 2.5" better still) and 3/8" vents and 1" thick. Three vents as a min. Angled backwards may not be a bad idea either.
If the rifle always bucks you off target, your ability to effectively LR hunt ALONE is greatly reduced. you can forget about the quick follow up shot because you will be busy trying to find the animal after coming down from recoil. In a few seconds, an animal that far away can move or duck behind anything and be lost. Now try and find it to track. Bad idea...
I want to see that bullet arrive, its affect, and where that animal moves to. Recovering the game is as important, more important, then the shot.
That is the main reason I suggested the fast 7mm and 30's. Way easier to tame that recoil and control that muzzle.
Whichever route you take, good luck and there is plenty of help here. Most here have owned most if not all the common LR boomers so have first hand experience of their requirements and limitations.
Ever consider the 270AM and the super heavy wildcat bullets? Another very effective option. See Fiftydrivers test drive in another post. I like that rig. Very useable parameters and would be right up your alley for type of rifle to fill your task. However, if you feel that heavy 7's to small, forget the 270.
When at the gunshow, consider a Savage or Stevens 110. They are great actions which can be built into rifles every bit as accurate as a Rem700 for 25% less. Bonus is there is an off the shelf long magnum mag box that can be swapped in to feed the RUM or Wby family of cartridges. If you buy one chambered in a RUM, bonus.
Good luck and shoot safe...