The 375 with good bullets from 230-270 grns is a wonderful choice, and one you would never regret. You need to ask yourself if all you had between a filled tag and an empty bag was a "rump to" shot on a good bull at 250 yds, would you turn it down? With the 308 or 6.5 you most certainly should walk away, but with the 375 you can take the shot with complete confidence that you will kill him forth with. All this blather about putting a little bullet in the heart or through both lungs assumes that the elk will give you a nice standing broadside shot........good luck with that!!! I have taken many, many animals of elk size and larger right out to 350 mtrs with a 375 H&H and 270 gn TSXs, of all these different critters I doubt 2 were standing broadside, some were hard quartering to or away, some were walking, some were running, some were behind a screen of bush, one all I could see was the top of his shoulders, my Bongo at 270 mtrs was in the bush/grass and all I could see was his head and neck down to about his back line. I slid a bullet along side his neck and into his spine between his shoulder blades....I severely doubt this shot would have been successful with a 308 or 6.5, given the amount of heavy muscle that was penetrated on the way to the spine.
There is absolutely no downside to using your 375, unless you are afraid of it. My rifle loves the 270 TSX but my son's Ruger loves the 260 AB and he took a dozen animals in Africa with that bullet ranging in size from bushbuck to eland, and believe me all African animals are tenacious and require a lot of killing, just like our elk. I highly recommend either of these bullets given the performance I have seen on more than 3 dozen head of big game.