I am going on my first Elk hunt this year in CO. I therefore will not comment on what caliber kills best. However, after hunting other game and taking care of gunshot victims in the OR, my relative experience is that it is all about shot placement. Shoot what you can shoot accurately and limit your yardage to your experience level, bullet energy, and bullet velocity needed to perform well.
I can speak with some authority on suppressors. I will second the TBAC (Thunder Beast Arms Corp) recommendation. I own their titanium 30p-1 for 30 cal up to 300 win mag use. I also hate the noise/muzzle blast of rifles. The noise suppression and recoil reduction of a suppressor is very impressive for both target shooting and hunting use (supersonic large rifle ammo now sounds like a .22LR report). When the bullet hits the animal it is a loud SLAP that I have not noticed when shooting unsuppressed. You can order a TBAC can from your local Class III dealer. You also need to ensure the suppressed hunting laws of the state in which you wish to hunt. It is becoming more common every year and for good reason, it protects your hearing and makes your rifle easier to shoot!
I have also shot the TBAC 338 model suppressor on a Barrett MRAD (Thanks to Barrett for sponsoring a side stage at the Mammoth Snipers Challenge!). The noise suppression was similar to my 30 cal model. The recoil was a slow push and not objectionable at all. The experience has got me considering a 338 LM suppressed rifle as well. As others have said, a 338 build will probably be heavy compared to what you are used to shooting.
Since you are an empty nester looking for something for you and your wife to do together, I would suggest you both start shooting practical long range (some listed as sniper) shooting competitions! Specifically, the 2 person team competitions in which both team members shoot and then spot for one another. These are great fun and will teach you a ton about shooting in all sorts of improvised positions and under pressure. Being a good spotter is even more difficult than being a good shooter and just as important if you want to stretch it out! I also find spotting just as rewarding as shooting. When you get your buddy on the target with the second shot, after a first round miss, it is pretty exciting. You will also figure out what your yardage limit is for the conditions and the stability of your shooting position. Also, you will learn what your individual limit is without a spotter to call corrections. This distance will be considerably closer than what you might shoot with a spotter. All of this WILL make you a much better and thoughtful hunter because you have an appreciation of your actual abilities in the field. It will also teach you to aim small because you are often shooting at 0.5-2 moa targets.
For competition use, I would recommend a short action rifle in 260 Rem/6.5 creedmore/6.5x47/etc caliber but a 308 is also a good choice if you do not reload (reloading is also something fun to do and rewarding if you now have the time). 338 calibers are not allowed at most comps. However, if you learn to dope the wind and shoot the short action calibers well
, I anticipate the magnum will seem easy in comparison. From a ballistic stand point, the 338 LM will shoot flatter and drift less, especially at the high density altitude in which the elk live, if your cardiovascular shape is up to the task! This, however, does not mean you can buy the big magnum and overcome a lack of experience with bullet BC and MV. It still gets pushed around by the wind! It is not a laser. You will likely miss or maim if you do not have the long range experience from shooting a lot of rounds, even with the big boomer (or quiet pusher if suppressed
Good luck and have fun,