Kinda interesting build! I usually start with bullets then look at range and game goals to kinda guide. I would build around the 250 Berger, absolutely crushes elk, shoots unreal and cuts the recoil a lot. You would not need to max out a large case to keep this bullet really hitting, I'm normally a huge 338 RUM fan especially with the ADG brass BUT I would probably look at the 33 Nosler, the Lapua and Norma take a different class of receiver which adds weight all over the place.
You'll actually have to probably have to add weight and go with the lightest contour carbon barrel to get a good mount with the suppressor, a #4 steel might cut back enough at 22 inches to mount to, I know a #5 will and in 338 is actually light.
I like the all Ti Pierce, really cuts weight, could also go with Defiances lightest steel action and still be pretty good.
I would probably put a Manners EH-5 on it depending on the contour of the barrel, the thumbhole really put the felt recoil inline with your body and what the suppressor doesn't eat up will still be comfortable and on target.