If you decide to use a Birchwood Casey bluing agent, choose the "Perma Blue".
It is made for use on the low carbon steels that muzzleloaders are made from.
The "Super Blue" is made for use on the alloy steels modern high pressure guns have and it will produce a very streaky, mottled look on low carbon steel.
Birchwood Casey Plum Brown will produce a brown finish that is more wear resistant than the Instant Bluing solutions however on large parts like a gun barrel it is difficult to get a evenly browned finish.
Plum Brown requires heat to work so some source of heat that will heat the metal to 280 degrees F is required.
If the metal temperature drops to below 250 degrees F the solution will deposit a layer of copper which will stop all further progress.
If the metal temperature is above 300 degrees F the solution will instantly boil off leaving ugly splotches.
Plum Brown also requires at least 3 coats, washing with water and drying with paper towels between the coats.
Plum Brown gives off some rather nasty smells too so plan on doing this outside of the house.
If you have access to a humid chamber like a bathroom with a shower, a cold browning solution like Laurel Mountain Barrel Brown & Degreaser is the easiest way to create a very hard, very wear resistant easy to apply finish. It's also low cost.