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7.62x39 will fit, howevever, 7.62x51 will not.

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Such rifles and cartridges would be "instruments of destructive surgery" on an animal as large as an elk. Keep in mind that your ballistics are not as good as a 41 magnum at the muzzle, but with the heavy bullets it would catch up and pass a 41 mag at over 75 yds.
Elk are big and can be tough. I hunt them every year, and I have for many years. I have been a guide for hunters for them for 30 years. I have killed one with a bow and arrow in the past, and several with rifles. I have seen the results of many many others. Hit with precision with a bullet that won't come apart, you can kill elk (or any animal) with a much smaller cartridge than most men use, but you had better hunt as if you were hunting with a bow. Just hitting them on the shoulder is NOT going to do a reliable job with such shells as you are discussing here. You might get away with it, but you might not too.
I have had to track an elk once for 6 hours that was hit on the shoulder blade with a 7MM REMINGTON MAG with the wrong bullet (140 grain, at close range) I have tracked more of them that I can easily count that were hit with 7mm and 300 mags (when they were hit a bit too far back) and it's not a fun job. An elk with a liver shot, gut shot, or hit in just one lung, can go a LOT farther than a white tail, and you measure those jobs in MILES and HOURS, not minutes and yards) Also keep in mind that most of these tracking jobs go into places you would not want to walk if your were not trying to track something, let alone doing it bent over. They will go into the steepest and thickest cover they can find in the Rocky Mountains, and trust me, there is a LOT of such country in the Rockys
IF YOU ARE WILLING TO GO HOME WITHOUT AN ELK rather than even risk a questionable shot, I would say go ahead.
But most men reading this don't live in elk country, and can't just "come back tomorrow" for free, or at least for very little money.
If you are wanting to hunt elk with an AR-15, and you don't have limitless time, please do your self and the elk a big favor, and get a 458 SOCOM or a 50 Beowulf.
Or, get an AR-10 (and it's copies) size rifle in a 260, 7-08, 308, 338 Federal, and never look back. And you should be shooting very good bullets in the 260 and 7-08. I recomend Barnes X for such hunting and Nosler Partitions in the heavier weights too.
In the 308, use 180 grain bullets of solid construction. In the 338 fed, go with a 210 gr Nosler Partition or a 225 grain Partition.
The suppressed sub-sonic weapons are very cool, and they really have a special function, but elk hunting is not that function. Run at full power without the suppressor, they are better, but it's still a light hitter compared to a 270 Winchester, 308, 30-06 and so on. Think of them like you would an archers arrow, and you can do well, but it's not the weapon I would choose unless you just want that extra challenge.

I hunted elk 3 years ago with my 6.8 SPC I didn't get one) and I think the shell is too small for general elk hunting, but I walked away from 3 small herds of them because I could not get the shot I wanted. If you can do that too, then I'd say you are an ethical hunter, and I would not ask you to leave the rifle home
But if time and money prevent you from walking away from the only shot you'll get on a hunt of a lifetime, (and at the expense of several thousand dollars,) GET A 458 SOCOM or a 50 Beowulf! Or an AR-10 sized platform in at least a 308 and a 338 Fed would be better.
Or just use more conventional rifle in 270 or larger caliber. And use tough bullets!

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.450 BM may be the easiest and least expensive upper to get ($550). .50 Beo may be the hardest to get support for, as I know Alexander Arms had been having issue delivering ammo and sales have been tapering off.

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One guy said that a 7.62 will not mate with a standard 5.56 lower, but they another guy said 7.62 x 39 will mate up with a 5.56 lower... Which is correct?  I am hoping that the 7.62 x 39 will work because of the cheap and availability of the rounds.