Howdy
Congratulations on helping out with that fine piece of history.
If I may, I would like to warn you, please do not attempt to shoot it.
Your revolver is chambered for what became known as the 44 S&W American cartridge. Sometimes just called 44 American, sometimes just 44 S&W. There is more to it than just getting the correct bullet diameter. 44 American is an obsolete old cartridge that employed what is called a heeled bullet. A heeled bullet had a small diameter shank at the rear that was inserted into the cartridge case. The case was then crimped around the shank, or 'heel'. The entire cartridge was the same diameter, both the bullet and the case. That is why if you look into the chambers, you will not see a shoulder in the chamber. If you want to see a modern cartridge with a heeled bullet, pull the bullet out of any 22 rimfire ammo. 22s are the only ammo today that uses heeled bullets.
The problem with heeled bullets was the lubrication was on the outside of the bullet. It attracted a lot of dust and grit. When the Russian government contracted with S&W for a large order of revolvers, they specified they did not want a cartridge with a heeled bullet. S&W came up with a new cartridge called the 44 Russian. The two cartridges are not the same.
44 Russian was one of the first calibers developed with a modern type bullet. 44 Russian was developed with a bullet that had the same outer diameter as the inside diameter of the case. The lube was contained in grooves on the bullet, and they were contained completely inside the case when the cartridges were assembled. That is why there is no shoulder in the chambers of your revolver. The shoulder is there to make the transition between the case diameter and the bullet diameter. The case diameter and bullet diameter in your revolver are the same, so there is no shoulder in the chamber. There were other cartridges made in the 19th Century that also used heeled bullets, but they are all obsolete now. Almost all cartridges now use a bullet that is the same diameter as the inside of the case and is inside lubed.
Even if you could find some cases appropriate for 44 American, nobody makes them, and if you could find an appropriate heeled bullet, you still have to figure out how to crimp the cases around the bullet. Modern reloading dies are only designed to crimp the case with a bullet of a smaller diameter than the case. You cannot crimp a heeled case with a modern die. I have read of some shooters who developed some special dies so they could reload heeled bullets, but they made them themselves.
Lastly, that pistol should only be fired with real Black Powder. Forget cowboy ammo. Cowboy ammo is mostly loaded with Smokeless powder, and although the pressure is purposely kept low, this is only to meet the requirements of the lower velocities often desired in Cowboy shooting. Even the lightest Smokeless Cowboy loads will still develop a higher and sharper pressure spike than Black Powder. That old S&W was never designed to take such a pressure spike. The steel used at the time is not up to it.
I recommend against Hodgdon's Triple 7 too. It is what is called a Black Powder substitute in that it develops a lot of smoke when fired. But Triple 7 is roughly 15% more powerful than real Black Powder. There is no need to subject that old gun to over pressure as a full case of 777 might cause. There are other Black Powder substitutes that you might try, such as American Pioneer Powder's APP, but as I say, you will not be able to find components, or crimp the cartridges.
There is one brand of ammo that does load with Black Powder, but they do not load 44 American, only 44 Russian.
So. No commercial ammo available. Can't take Smokeless pressure. No commercial cases available. Have to find heeled bullets. Have to make your own crimping dies.
Please, enjoy that wonderful old revolver as the precious heirloom it is, don't attempt to shoot it.
By the way, I shoot a lot of Black Powder in both modern firearms and antiques. I have a New Model Number Three that was made in 1882. I do shoot it occasionally with Black Powder. But it is chambered for 44 Russian, cases are available, and I would never dream of shooting it with Smokeless powder.
Sorry for the long post, but I felt you needed to know.