In my 1899 Navy the cylinders are not bored through. A 38 special round will not go all the way in. It is about a 1/10" short of going in all the way. You can see in the cylinder it is not bored out.
Will the .358 lead bullets work with a light charge of bullseye or should I find .361 bullets and black powder?
Some of the information above contradicted what I could remember, so I pulled out some respectable resources, such as "Textbook of Pistols and Revolvers" by Julian Hatcher, and "Handloading Ammunition" by J. R. Mattern. I'm not going to print everything, mostly because the 2 year old "Godzilla Destroyer" is coming to visit...and I have to hide.
The .38 Long Colt was originally made to take a hollow based .357 diameter bullet. Bores were oversized and depended on the hollow base to expand to fit the bore. If you load a bullet that has a diameter large enough, it would have to be forced into the cylinder and would raise pressures quite a bit. Of course that would depend on the individual gun, but I would stick with the .358" bullets until you figure that out.
The original factory loads appeared to use Bullseye powder. I'm not going to give the charge because they don't give details about the type of case heads used, so I'm going to assume they were of a balloon type. And again, I would be worried about the heat treatment of your individual firearm and how much pressure it would take to do permanent damage to it. Maximum pressure, according to J.R. Mattern, was 12,000 lbs. I'd try to get as low under that as I could.
I shoot .38 LCs a lot, through a .38 special. I have a Lee die set, which didn't cost nearly as much as they appear to be now, but the dies as marked are: Sizer- "38-357", Expander- "38-S&W", Seat/Crimp- "38 Auto".