To clarify, the .38-44 Target cartridge has a case diameter of 0.383" and a length of 1.475". The bullet diameter is 0.361" (nominally the same as the .38 S&W bullet). There was also a shorter .38-44 gallery load which used a lead round ball bullet. There is no reason to not use .38 Special, .357 Mag, or .38 S&W brass. If using .38 Special or .357 brass, it might be best to leave them un-resized, as their OD is a little small. If you can find .357 Maximum brass, that would also work, but would require trimming. Regarding bullets, just as the .38 S&W is stated to use .361 bullets, I find that .358 lead bullets do work just as well, therefore, I'd just use any .358 lead bullet for the .38-44 if I couldn't find any in .361.
If I had a #3 in that caliber, I'd probably try mild .38 S&W loads to start with (asssuming that they would chamber OK), and if their performance is satisfactory, there is no need to go any further. It might be interesting to try .375 round balls (downsized a bit) in a .38 Special case. As a propellant, I would use a minimum load of Bullseye or similar, essentially the same amount as would be used in a .38 S&W load. You wouldn't want a MV to be much more than around 600 ft/sec, but enough to keep the bullet from sticking in the bore. You could certainly use BP, but that can get fairly messy, and it will gum up the works in short order.
