I've done a little research.
The old visitor’s center/cyclorama and the observation tower are “nestled against” Ziegler's Grove. Ziegler’s Grove is located between Taneytown Road and Hancock Avenue. The Angle is south (to the left from the Union perspective) of Ziegler’s Grove.
The point of Gen. Pickett’s Division's charge was directed just left of the Angle. Gen. Trimble's Division's charge was just to the right (from the Union perspective) of the Angle, and Gen. Pettigrew's Division's charge was far to the left of the Angle. I *think* that the only two regiments under Pickett to reach the batteries was the 57th Va. Inf. and Lowe’s 28th NC Infantry Regiment. If that is correct the sword seems to have come from an officer of the 57th Va. or the 28th NC…or was possibly even that of Brigadier Gen. Lewis Armistead.
Gen. Armistead is known to have been waving his sword and shouting when he was killed. Armistead was shot three times just after crossing the wall. But, he did not die on the field. He died in a hospital two days later, and surely the Union soldier would have known if he had gotten the sword off of the general.
A sword with a South Carolina palmetto would seem more likely to have come from a NC soldier. And perhaps it was not a SC palmetto, but the NC pine tree…often mistaken for the similar SC palmetto. Much of Lowe's 28th NC was made up of men living close to the SC/NC border. Some of the NC soldiers lived in parts of NC that had formerly been in SC before border changes moved the Stateline. It seems to me most likely to have come from that regiment.
At any rate, it must have either been carried by Gen. Armistead, a soldier of the 57th Va., or a soldier of the 28th NC. My ancestor, Col. Saml. Lowe, was leading the 28th NC, but he was "only" wounded. He was not killed, so it was not his sword.