Hello Bekerb
My name is Steve Zihn. I am a gunsmith. I earn my living making muzzleloaders, but I have done gunsmithing on many types of arms through out my years.
My 1st passion is flintlocks, but I love any kind of shooting that burns powder.
I have a considerable amount of experience building competition center fire rifles and sniper rifles in years past, and also I am familiar with the high accuracy world of the AR-15 rifles.
The reason I tell you all this is to illustrate that I am not a new comer to any of this.
The idea that a sling swivel mount on the barrel is going to harm accuracy is not necessarily true. Look at the world of 100 yd competition and the before mentioned AR-15 futurity. Many if not most of these rifle have muzzle breaks on them. If you have followed "high Power competition" any at all in the last 6-8 years, I am sure you know that there are NO 30 caliber records anymore. ALL are now held by the 22s 24s and 6.5MM. And I am talking about 1000 yd matches too.

You see, hanging something on the barrel is not going to harm your accuracy as long as it's not touching anything. In some cases (like the Browning BOSS and it's copies) it will actually improve the accuracy.
I Have 4 scoped rifles with sling mounts on the barrels, a 25-06, a 270 winchester a 338-06 and a 375 H&H.
My 338-06 is the least accurate and will "only" group at MOA now and then, but mostly I can keep them into about 1.25" at 100 yds. ( I have never seen an elk that small)
The other 3 rifles all shoot under the M.O.A. mark, and the 375 and 25-06 are shockingly accurate. They shoot UNDER MOA.
The trick is you can't have any pressure on the sling when you are shooting. It needs to just "hang" down so it's not effecting anything one time and not the next.

So you need not worry about loosing accuracy on your "classic" rifles as long as you understand them. As far as having a barrel mounted sling on a heavy hitter (416 or larger) I would not care one way or the other if I had "only a 2 inch gun".
You see, you'll never be shooting so far at something small enough to would make any difference at all with such calibers.

Happy hunting
Steve