Howdy
For what it's worth, my 45-70 Pedersoli Sharps also came with a little red card inside the box saying Black Powder Only. Never use Smokeless it says. I have never fired any Smokeless through it, nor do I have any desire to. But the falling block action of my Sharps is a much stronger design than my Trapdoor, and the barrel is much, much thicker, so I think at least in this instance at least Pedersoli is following the company lawyer's advice and covering their butts. I don't say the same thing about old Colts and such, but I do suspect this is the case with Pedersoli. In short, I would not hesitate to fire moderate Smokeless loads through my Sharps if I wanted to.
Regarding Smokeless Trap Door loads, my 13th Edition Speer manual has three separate categories of Smokeless loads for 45-70. The lowest power loads are the Trap Door loads. Specifically, the loads listed in this section are for "original trapdoor Springfields, modern trapdoor replicas, original Sharps' Model 1874, and rolling block actions." In the text it goes on to further list the cautions about shooting these loads in original Trapdoors, including a thorough inspection by a gunsmith familiar with the actions.
Speer states that current (as of 1998 anyway) factory pressure limits for modern factory 45-70 Smokeless ammunition is 28,000 cup. However the loads in the Trap Door section are kept to a maximum of 21,000 cup. It lists four different bullets in this section, a 350 grain Flat Jacketed Soft Point, a 400 grain Flat Jacketed Soft Point, a 325 grain cast 'cowboy load', and 500 grain cast bullet.
The other two sections for 45-70 in Speer are the 'lever action' loads, and the Bolt Action and Modern Single Shots loads.
The Lever Action loads section specifically states the loads are safe in "newly manufactured Marlin Model 1895 rifles, Browning Model 1886 rifles, replica (not original) Sharps Model 1874 single shots, and original Winchester Model 1886 lever action and Model 1885 single shot rifles know to be in good condition (see text)." The pressure levels of the loads in this section are held to 28,000 cup.
The hot stuff in the Bolt Actions and Modern Single Shot section is specifically ONLY for the Ruger No. 1 and No. 3, the Browning Model 1885 single shot, and M98 Mausers properly converted to 45-70 ammunition". This stuff is held to a maximum of 35,000 cup.
So, I can't imagine why you would want to put Smokeless powder through your new carbine, but at least according to Speer, you should be good with loads up to 21,000 cup. I dunno if I would want to experience that recoil with a carbine though.
As an aside, a number of years ago, before I started loading my own ammo, a friend loaned me a modern Marlin carbine chambered for 45-70, just to fool around with a little bit. I don't remember specifically what the ammo was that I bought to shoot in it, but I can tell you that from the bench it hurt so much that I gave him back his rifle and a whole lot of unshot ammo.