I think David McKay Brown makes about the best looking 600NE available.
Much of the aesthetic quality is from using a round action - so that answers one question. So many other 600s are so big and disproportionate that I think a round action helps that a lot.
Also, a gun of such stature should really be housed in a sturdy oak and leather case simply from a durability standpoint but that does add some extra weight in the overall package.
I would have sling swivels simply because any added stability as proper use of a sling gives is a plus, not to mention that if you were to carry such a rifle, a sling would be a welcome addition.
As for the extra barrels, I am thinking that such a rifle would have an action sized for at least a 10bore if shotgun barrels were wanted. If a second set of rifle barrels is considered, maybe a lighter caliber with Daniel Fraser style barrels, that is, barrels with a heavier breech end (to fit the action properly) with maybe 4 inches or so of slight taper to a pronounced step to a lighter profile. But, what lighter caliber would be worthwhile on a 600 sized frame that isn't still a stopper class cartridge? IMO, once a 600, that's pretty much what it is because it is such a big step from 500 based cartridges.