Thank you everyone for your feedback.
I own quite a few Civil War fired Whitworths. This bullet has the same shape, dimensions and look of a cylindrical Whitworth. I bought it a while ago from a reputable and experienced dealer who certified it was one. I also have a signed and detailed statement of its provenance which I have no reason to doubt.
I'm aware of the reference in the Thomas & Thomas Handbook (page 25). The authors say that Whitworth-like long bullets with dished bases and a dot in the center are actually .45 caliber Sharps bullets from the post-Civil War period. I have another bullet, which is a guaranteed hex Whitworth dropped at Spotsylvania. It too has a dished base, but no dot in the center. It's the dot in the center that looks different to me.
The pictured fired bullet was found (legally) in the area between the saddle of the Round Tops and the Taneytown Road. I read that on July 2 a small group of Union soldiers, including Berdan Sharpshooters armed with Sharps rifles, took up a defensive position (facing westwards) behind a north/south running stone wall. The stone wall was east of Little Round Top and the saddle (between the Round Tops) and was on the western edge of a field that bordered the Taneytown Road. The position was about 150 yards from the advancing 15th Alabama moving towards the southern spur of Little Round Top. This is in the vicinity where the relic was dug. If that was the case, then the bullet could not be a Union Sharps, as these Northern snipers used larger caliber Sharps bullets with grooves/rings.
If the discharged projectile was dug in the area at the rear of the Federal manned stonewall, it's more likely to be a Confederate fired bullet.
If it was a Confederate Whitworth, I couldn't find any evidence of Whitworth sharpshooters being present at the southern end of the battlefield. Please see my thread, 'Whitworths fired at Little Round Top' recently posted under the Battlefield Forum, 'Gettysburg'.
Although the bullet appears to be a Whitworth (except for the dot in the center of the dished base), with no account of Whitworth activity in the area, I am now questioning what kind it is. If it's not a Whitworth (or Sharps), could it be another kind of bullet maybe fired by a Confederate sharpshooter with a private target rifle in one of the Texas or Alabama regiments approaching the southern end of Little Round Top?
Any further comments, thoughts or ideas would be welcome.