I have a replica 1859 Armi Sport Sharps, but I have no knowledge of the originals so can't help you there.
Missionary is correct, the most straight forward path is probably to shoot it with loose powder.
But making the paper cartridges is pretty easy. There are a lot of Youtube videos and websites that show you how to go about it. You can also buy Charlie Hahn paper tubes that work with the Sharps Christmas tree bullet molds. I just use an ordinary wooden dowel as my forming mandrel, and I wrapped it with that Air Conditioning tape (the shiny smooth stuff) to get the OD correct. Drop your bullet in the chamber, push a wooden down down on top of it and mark the dowel with a black sharpie pen at the breech to get the OAL for the cartridge. It is all pretty straight forward.
I just got my hands on the Era's Gone .54 caliber Sharps mold last month as a birthday present. I haven't had a chance to cast with it yet, but I am looking forward to it.
I have 3 different Christmas tree molds - a Moose Mold, a Tennessee Bullet Molds and a Rapine. I also have the Lee Hollow base .54 415 grain Minie (90474) and the .54 caliber 300 grain REAL (90398) bullet. I have shot them all with paper cartridges and the Christmas tree bullets with Charlie Hahn tubes.
As far as sizing and lubing, I picked up a .557 sizing die for my Star that I keep loaded with SPG. I don't worry about sizing them, but filling the lube grooves with the Star uses far less lube than dipping them. And so far, it seems to be enough lube in my carbine length barrel.
I recently bought a Lee .562 round ball mold to play with. I loaded them over the Charlie Hahn cardboard tubes. I haven't had a chance to try them yet, work has been too busy. Here is a thread on this site if you may be interested.
https://www.artfulbullet.com/index.php?threads/1859-armi-sport-sharps-54-round-ball-loads.8562/
Here is my carbine.
These are some cartridges I loaded with both homemade paper cartridges and the Charlie Hahn tubes. The ugly one on the bottom is from my first try back in 2014 using lady's hair curling paper tying it to the Christmas tree bullet base and dipping the bullet in SPG. The top two are the Lee REAL and the Minie in my paper cartridges. They all shot well, but I was partial to the results I got with my Tennessee Christmas tree mold which are the center ones in the cardboard tubes. Using the Hahn tubes with the Christmas tree bullet is a cakewalk and easy to throw them together.
This was an experiment where I left the lubed bullets in the Houston sun all day in the heat of a summer day. The lube simply stayed put. The left bullet is the Moose Mold Christmas tree bullet with the rebated base that is slotted to accept twine to tie it off to the paper cartridge. The right bullet is the Tennessee Christmas tree mold that is also rebated, but designed to slip down into the cardboard tubes. You can see what I meant by the quantity of lube on the bullet when dipping them versus sending them through the Star lubesizer.
And the MTM 20 gauge utility box in the photo above is the perfect size for holding paper cartridges. I originally tried to use the blue plastic ammo box shown, but I found I would rip the cartridges open when trying to pull them out. The MTM box works much better and there is room inside for your caps, cleaning jags and other things. It was less than $20 at my local Walmart.
Good luck on your project. I was really surprised how much fun these rifles are to shoot. If you find out you need .54 caliber projectiles, let me know and I can send you some to try.