I have a 21" 338 Sherman Mega and I am using a Rugged Alaskan can on it. It is still pretty loud when compared to my Dead Air Nomad on my 21" 300 WM, but it is more than enough recoil reduction and muzzle blast reduction for hunting purposes.
If you are looking for suppressor recommendations, the Rugged Alaskan, Griffin Sportsman Ultra Light 338, the AB Raptor 8, and the SilencerCo Omega 36M (Short config) are all good suppressors in the less than 15 oz range. If you want to go a bit heavier (but probably get better suppression), the TBAC 338 Ultra, the Dead Air Nomax 33, and the YHM Bad Larry are good, but they are in the 17 oz range.
Ultimately, you have to decide if you want the best suppression or the least amount of weight. A short magnum cartridge is still going to be "loud" when compared to a long barreled standard cartridge using the same suppressor. Today's suppressors are much better than the ones from even 10 years ago, but you have to be realistic.