We're getting there. But a little more information is needed.
You inherited a .22 Caliber blued S&W revolver with a 6-inch barrel stamped .22 Long Rifle CTG and it has fixed sights. Originally the S/N indicated was 253813, however you say there is a "B" prefix.
We have to discount the "B" prefix for two reasons. First you say it doesn't appear on the butt of the gun. Second is that a "B" prefix would be for a small .22 semiauto pistol, not a revolver. The prefix would be obvious on the butt of the gun, as shown here on a different revolver.
Again, photographs would help if you have them. If you have the original S&W box, any wording on the box would be a clue, especially from the end flap of the box.
Below is an example of a fixed-sight S&W and it should look similar to yours.
Note the rear sight is a fixed "square notch". And the barrel is
round with no rib on top. Does yours look different? Does the gun have a round barrel or a narrow grooved rib on top of the barrel?
More questions:
- Does the cylinder have a 6-shot capacity?
- Is the back of the metal grip frame smooth or serrated(grooved)?
- Same question about the front of the grip frame.
- Compare your front sight to the photo above. Is yours more rounded? The same? Square at the rear of the blade?
- On which side of the frame is the round S&W logo?
- Is the logo large (about the size of a nickel) or small (dime sized)?
- Is there a screw at the top of the sideplate (right side of frame) just below the rear sight?
- Is there a small screw on the front of the triggerguard?
- Last, and this will sound odd; can you provide the length of the cylinder in inches? Precision is good, but "close enough" would be just under 1", about 1.25", about 1.5" or a hair over 1.5".
Depending on the answers to these questions we might be able to identify your revolver.
In any case, I'm leaning towards one of two valuable firearms at this point.
The K-22 Outdoorsman
The M&P .22 Caliber ("Post Office" gun)
Most .22 caliber guns, even the small framed guns, have adjustable sights. These are identified by the rear sight sticking up above the top of the frame with screw adjustments at the top rear and side. If the sight has been removed, there will usually be two empty screw holes on top of the frame and a flat horizontal (L/R) notch at the top rear of the frame.
If there is a small "hump" on top of the frame just at the rear sight, your gun should look similar to this nickle specimen.
The 7-shot, M-Frame .22 Long Ladysmith