Hunt with the .338ME. Don't worry about bears and such.
I've probably hunted where you are talking about. The Grizzly's of Wyoming, Montana, Idaho aren't the same critter as the Alaskan coatal browns. (Technically, they are Ursus horribilis, but the coastal browns run to upward of 1,000lbs. The grizzly of the lower 48 typically don't top 500-600lbs). We have black bears in Georgia (Ursus americanus) that go that big as well as feral pigs. The state record for the black bear in Georgia is 733lbs and was killed by a single shot from a .243 with 100gr Remington Corlokt. Folks are chasing them as I type this with bows and arrows......
The 200gr FTX will do fine on the elk. Why it "works" on a 700lb elk, but not on a 600lb bear at much, much closer range is another discussion for another day...
I hunted with confidence in Colorado last year with my .338MX. It was a relief as it's 3lbs lighter than my .300RUM. After the first day of lugging the .300RUM for over 10mi at 8-10,000', I really appreciated the Marlin on subsequent days. If surprised by a bear, I most likely could pump 5rds from the MX in the time I could get off 2 from the .300. I'm very confident with my MX. I've taken several deer with it and all have been bang-flops and bullet performance lacks for nothing.
I've worked up a load with the 250gr Hornady RN-Interlok. It will certainly do the job on the bears. I took 5 in case I had to hike back in to pack out meat, in case I ran into "smokey". Where I normally hunt in Wyoming, they have been known to claim carcasses. Yeah, if I KNEW I would face an irate bear, I'd take the .375Ruger with 300gr Nosler's, but I'd still strongly consider the "fire power" of the lever action Marlin. I don't think I'd take the .45/70 GuideGun with 400gr Hard-Cast slugs over the .338 with the 250's... But, I've recovered some of the 250's from an oak stump and "like" what I saw... I'm not sure the 220gr RNGC from the .338 at 2,100fps would stop in a 500lb bear. I've yet to recover one except from my berm, where they hold together very well.
I can't see how the handgun with a lead bullet would be preferable to the .338MX with a 250gr controlled expansion slug at 2,250fps (my loads).
Yes, I have a S&W 329PD and a New Model Ruger .45LC. And 300+gr bullet molds.
I'll take the .338 Marlin. (I have two of those too, btw a JM .338MX and Remlin .338MXLR).