Roy, You know your guns, I think you made a fantastic choice with the 375 AM and the 338 ultramag. I am jealous and want one also for extreme long range. Good luck with it. If I drive to Idaho would you let me shoot it? It would probably be worth the trip.
I built and tested quite a few of the 338-300 ultramags (338 edge on this forum) back in the late 90's before the 338 ultramag came out. After testing about a hundred rifles there is virtually no difference in performance between the 338 RUM and the 338-300 ultramag. The 338 RUM is the better cartridge between the two because of the slightly shorter more efficient case allowing better feeding through the action with longer high BC bullets. Your 2750 fps 300 grain load with your 338 ultramag in a 26" barrel is all any average 338-300 or edge will do. With a 30" barrel both can get 2800-2850 fps max unless it is a very rare, unusually fast barrel. After the 338 ultramag came out I never built another 338-300 because it made no sense. I kept two 338-300's for myself and hunt more with a 338 RUM. If I want more power I go to my 338-378 wby with way more power than anything off the ultramag case. All are very accurate if the gunsmith knows what he is doing. The 338-378 I hunt with is a solid 200 fps faster than the 338-300 ultramags with single digit velocity spreads between shots and extremely accurate. The last kill with it last month was an antelope right through the heart at 813 yards and it was not a very difficult shot with that rifle. 200 grain nosler at 3640 fps.
Time to go bigger though for extreme long range so I really like your 375. I get bored with this 1000 yard stuff now. My 338-378 with the 300 matchking right at 3100 fps shoots real well and I have made some good hits to mid 1300's with it. Your 375 can go beyond a mile though and that is where I want to go next. Best of luck with it.