Yikes, Brent, those bad boys will knock the squeal out of the biggest pig.
Mag Wheel Man...
A point to ponder: It's forces that cause mechanical things to fail, not pressure per se. Pressure is just a way of keeping track of the numbers.
What you want to know is the force the black powder cartridge will apply to the SxS's action as compared to the force of a shot shell. Every engineering student can tell you the universal F = PA (Force = Pressure times Area). To get the force in pounds, multiply the pressure (in pounds per square inch) by the reaction area -- the bore area, for the sake of demonstration (in square inches).
Little pressure data is readily available for .577 Snider. So let's use a .577/450 Martini-Henry in an example:
Pressure = 25,400 psi (from reloading data). Area (Pi (r squared)) = 3.1416*(0.45/2)**2 = 0.16 sq in
F (.577/450 M-H) = 25,400*0.16 = 4,064 pounds.
For a typical 12-ga load:
Lets assume a pressure of 12,000 and an area of 3.1416*(0.729/2)**2 = 0.42 sq in
F (Typical 12-ga) = 12,000*0.42 = 5,040 pounds.
With your proposed project, it seems the breech force is not a problem with any modern SxS. You need to be concerned about the strength of the barrel liners. The rifle cartridge operates at a higher pressure, but it's smaller cross section results in less breech force than the lower pressure shot shell. The "Military Load" for the .577 Snider is a 480 grain lead bullet delivered from the muzzle at 1,250 fps. Which is just a little less than a heavy target load of 493 grains of shot at the same velocity.
[SIZE=-2]Note: This example is based on assumed maximum pressures for demonstration purposes only, and should not be used as design criteria.[/SIZE]