IMO, if you want the best accuracy, you're going to have to look for a suspended barrel. That means one of the S&W models with this feature such as the, 620, M&P R8, 386 Light Hunter, or the 327 TRR8.

If you want to look for another brand, try and locate a Dan Wesson revolver, when properly set up and in capable hands, they have a reputation for being capable of 1/2 inch groups at 50 yards. As for why a suspended barrel, in this system the barrel is restrained at two points, the frame at the rear and the end of the barrel at the end of the barrel shroud. This greatly reduces barrel whip due to a bullet transitting the barrel. One plus for the Dan Wesson as opposed to the S&W is that the lock for the cylinder on a Dan Wesson is at the crane, moving the lock point as close to the B/C gap as can be achieved. Which is why S&W has toyed with a detent in the crane on some models over the years. Unfortunately, Dan Wesson revolvers are no longer made, so all you can find is used.

I have the 620 which is fitted with a J Point reflex and from a sandbag rest I've shot a 1.75 inch group at 35 yards with Speer Lawman 125 grain 38 spl. TMJ. This group featured a vertical deviation of 3/8 inch and a horizontal spread of 1.75 inch. I have no doubt that if my release skills were better I could have grouped it under an inch, probably in the range of 5/8 to 7/8 inch. Point is, even with 38 spl. with the right ammo this gun can shoot. BTW, with cheap Federal American Eagle it'll group at 2.25 inch at 35 yards with the spread being mainly vertical. It's a darned shame that Speer stopped making the 125 grain TMJ, I shot my last box getting that 1.75 inch group so now it's back to looking for something else as good. Next up to try is the current Lawman TMJ, a 158 grain in 38 +P.

Now, if you want to use one of the more common models, such as a 686, I would suggest that you go a bit against the common wisdom. From an Engineering standpoint a SHORTER barrel will have less "whip" due to the bullet transitting the barrel. Since optical sights are now permitted in bullseye shooting, there really isn't any advantage to a longer barrel. Reflex sights produce a projected image at a distance equivalent to 40 to 75 yards, so in practice the sight radius equivalent to the distance to the target. So, if you're shooting at 50 yards, you have an effective sight radius of 50 yards. That pretty much negates the advantage a long barrel provides and allows you to use a shorter barrel that is effectively much stiffer. For a 686, my recomendation would be the 3 inch version simply because it will insure a full ejection stroke, however for a pure bullseye gun the 2.5 inch would have a slight advantage in terms of barrel whip.

As for the "bullet jump" when using a 38 spl. in a 357 Magnum chamber, my own experience is that the effect on accuracy is not measurable unless you've got the gun bolted into a machine rest. For me, the 38 spl. is far more accurate because I can't afford to shoot enough 357 Magnum to get my recoil control to the point where I can shoot it well enough for precision. At 35 yards my group sizes nearly triple with the Magnum and it's all due to somewhat poor follow through. I also can't group well at 50 yards because these old eyes don't see the target well enough at that range, however I do have some fun at ranges where I can still see the bullseye.