Are you limiting it to those four and just one rifle? If so, here's my take. I've had or have a 375 H&H, 375 Ruger, 375 Mashburn, and my 416 was a Remington; my .338s included both the Winchester and the 340 B. I've not taken a brown bear but have been around them enough to have at least a fair idea of what I'd be comfortable carrying for them (and I was comfortable with the 340 and 250-gr Nos Parts).

With the first three in question, the platform would be the most important to me as all are capable for your noted goals. I don't think you'd need the 416; or at least, it'd be my fourth choice as an all-rounder. As to the 338s I've had, including the 340, I find them just as disruptive on the shoulder generally speaking as the 375s mentioned. Plus, now with the 260-gr AB and the 250-gr TTSX 375s you give up nothing in trajectory to the 338s.

For me then, it'd be one of the 375s, unless of course, as mentioned you come upon the 338 platform that snuggles up to you just right.

But then, other than these four, there are the 9.3x62, the 30/06 with 200-gr Nos Part's, and then some others..