I have an original Remington Hepburn in .40-70 SS.  I have shot this rifle for 40 years. The groove diameter is .410 or so. I can't remember exactly. It has the early 5 groove barrel so the groove diameter can be a little hard to measure if you don't have experience with the odd numbered grooves. I shoot a .410 grease groove bullet(between 300 and 390 grains) and 21.5 grains of IMR 4227 breech seated. It will shoot 243's on the standard ASSRA 200 yard target. If you want to shoot fixed ammo you will need to use black powder (or duplex) and paper patched bullets which are .403-.404. The brass is available from Buffalo arms or easily made from .405 Winchester brass and it comes out at the right size. You will actually need to trim the .405 brass since it is a little longer then 2.5 inches. You will not be able to shoot groove diameter bullets from the case since it is chambered for bore diameter paper patched bullets.. The reason you think that the rifle is  .403 groove diameter is because Cartridges of the World says it takes a .403 bullet. What they don't say is that it is the paper patched bullet size from the time when its cartridges were loaded with black powder. Remember it is "Cartridges" of the World Not "Groove diameters" of the World.
P.S. You might want to use rifle primers instead of pistol primers due to the Hepburn's rebounding hammer.

The deal about the primers is: the Hepburn's rebounding hammer leaves the primer unsupported by the firing pin while there is still some pressure in the barrel. This makes it want to force the primer back into the firing pin hole;thus making it hard to push down the block and extract the fired case. The rifle primer is just stiff enough to resist this flow back into the firing pin hole. 
The .405 case will fit the rim size and thickness much better then the 9.3(which I have tried) The base and rim size of the .405 fit my rifle perfectly.
All that is required for the 405 to become a .40-70 SS is to run it through a .40-70SS (original size) resizer and trim it to 2.5 inches L.O.A.  You may want to neck turn for a specific bullet which is a little bigger then .403 but I would not advise it. When I tried it after the shot I ejected the case and it was only about 2 " long. The thinned neck portion of the case stayed with the bullet and went all of the way through the barrel. In essence I shot a jacketed bullet. 
Let me know if I can answer any other questions. As far as misfires go, I don't have any problem unless I use cheap primers. Stick to Winchester or Remington. It's also possible that you need to take out the firing pin and clean the firing pin channel.

I also use .405 brass in my Hepburn, but you might find the .40-70SS dies wont quite bring the case down to size just ahead of the rim enough to allow easy chambering. I have found that either turning the last bit in my lathe, or running the cases into the die without a shell holder, and then pushing them out, will get that last bit to size down.

Consider: get your reloading dies from CH-4D and specify that you want a .401-.402" expander.  they will supply any size expander that you want.  the other dies are interchangeable.

Or, If you've already purchased a set of dies, just get the custom expander die from CH, again specifying bullet size and overall case length.  Cost is reasonable, and beats buying a second entire set of dies by a long shot.  They have a Universal Expander body and a large stock of varying diameter expanders that screw into the body.  Should be no back order.

I have an early 3-digit SN Rem.- Hepburn .40 2 1/2 with a nominal .401" groove dia. Slug actually mic.'s .4075" groove dia. Folks call this a .40-70 SS, but I understand that Remington called their version of this cartridge a .40-65 - at least according to Grant.

I've had the rifle since 1974 and have used all sorts of brass in it since then - original SH cases, berdan-primed cases, reformed 9.3, trimmed .405, and stretched .30-40 cases. It came to me with a batch of brass that is now long-gone and its own Remington grooved mould and bullet seater, so I've never tried any other bullets and never bought any dies for a reloading press. 

As the original brass wore out, I found that trimmed .405 brass would work, but was too thick at the neck to allow my grooved bullet to chamber. I would run the cartridge ~3/8" into the carbide base sizing ring of a Lee .40 S&W Factory Crimp Die - which would slightly reduce the diameter of the cartridge just enough at the neck to allow it to chamber freely. No-doubt...this process swages the bullet and likely affects accuracy - but it allowed me to shoot the rifle without inside-reaming the necks of the .405 brass.

In 2006, I discovered that Buffalo Arms had the capability to stretch .30-40 Krag cases to the proper .40-70 SS length and the other dimensions for my particular Remington-Hepburn...and swage the case rims to be slightly thicker to properly match the headspace requirements.  The necks of the stretched cases are drawn thin enough to receive the proper-size grooved bullet without employing that damaging Lee FCD swage procedure. My Hepburn is now standing in tall cotton with respect to case availability, and I have shot it more than I ever did when I first acquired it.

Call or email Buffalo Arms for information on their line of stretched Krag cases for .40-70 SS with respect to your particular headspace requirements.

(You need to Login or Register to view media files and links)