The original .45/120/550 Sharps was the 2 7/8" case.

Sharps never chambered a 3 1/4" case in a Sharps rifle.

In modern times, writers called the .45 3 1/4", a .45-120-500 Sharps, or short formed, .45 120 Sharps.
Thus today, most all people call the .45 3 1/4", a .45-120 Sharps or a .45-120-500 or 550 Sharps depending on the bullet used.
In a modern action, there is no reason why the .45-120 "Sharps" cases currently available could not be loaded to the same ballistics as the .450 Nitro.
I used 116gr. 2f GOEX in RCBS cases and 122gr.2F GOEX in Bell cases.
Bell cases in .50 3 1/4", easily held 140gr. 2f GOEX AND 170gr. 1F Meteor (made in Scotland). The meteor powder was much heavier per volume, than GOEX or C&H of the day.
In a Shiloh Sharps rifle, the .45 3-1/4" case can be loaded to just over 1,800fps with a 500gr. cast bullet. The load listed in Hodgdon's annual manual this year, is 60gr. H4895.
There are 3 other loads listed that exceed 1,700fps with the same bullet and producing under 29,000CUP.
Incidentally, their data for the .50 3 1/4" case they call the .50-140 Sharps, lists all 3 loads with 425gr. bullet at over 2,200fps and all 3 of the 515gr. loads at over 2,000fps are under
28,000CUP, thus safe in the Sharps replicas.
Much depends on firing pin diameter. Black powder pins might start to leak primers at over 26,000CUP - caution.
My bro's new .45/70 Sharps from Shiloh or C Sharps Arms???- had a modern smokeless small diameter firing pin, so it should be quite fine up to whatever pressure the maker notes -
which I do not know at this time. Hodgdon supplies data, though.

Further to this, 30 years ago Accurate Arms powder mfg'r's data had 500gr. jacketed running just about 2,150fps using AA3100 powder in the .50 3 1/4" "Sharps" case. IIRC, the load was 103gr., however do not trust my memory on that.
That is the load my late buddy Brad used in his Shiloh Arms Sharps. It was a kicker, with the hooked steel butt plate.