I haven't shot a Sharps & Hankins. The Owziak 56-56 bullet works well in both Spencer and Joslyn. It is available in 0.535" and 0.542". It does not shoot worth a hoot in my Ballard rifle. Although chambered for a nominal 56-56 cartridge, the bore is actually 54 caliber. There are some boxes of original 56-56 ammo with oversize bullets marked "For Ballard", so mine probably isn't the only one. With a 0.558" bullet, it shot quite well at the last N-SSA National. The only problem is that the chamber has no taper at all. If the cases are a little hard, you get some gas back through the action, which is mostly a cleaning problem. If you anneal the cases well, they can stick in the chamber. I needed to rod out a few cases on the line.
Your modified Lyman may work OK even if it is undersize, since it is hollow base.