MEHAVEY, I've been shooting BP for over 40 years, and have yet to find any rust in the bores of any of my rifles, shotguns or handguns., That is why you mix the ingredients with equal parts of Alcohol, Hydrogen Peroxide, and Murphy's Soap Oil, after using this solution, I use a healthy dose of Thompson's Bore Butter. Tell me have you even tried it?
H2O2 certainly can cause corrosion. It is an oxidizing agent, and isn't processed in carbon steel because of that corrosion.
I've been shooting muzzleloaders a bit myself, and I have tried the recipe. One of my best friends was a chemist, and thought the concoction rather funny since the H2O2 will oxidize quickly, particularly if the container is not light proof, and the soap and alcohol provide chemical neighbors to help the H2O2 become water. H2O2, in weak concentrations, will react with the powder residue to render it colorless, but the removal happens with the soap.
Bore butter works well for you and me, being in the dry southwest. I don't use a lot, as it doesn't take much. But moving from Oregon, I find rust to be an endangered species here. Back home (in western Oregon, or before that, Tennessee) rust is a real bugaboo, and I used a light coat of RIG instead (Tennessee was danged near impossible, what didn't rust would mildew). Don't know what a "healthy dose" amounts to, but I find a light coat keeps the rust away, and I don't have ignition issues due to over-coating the barrel. Works for both my firelocks and my percussion rifles and guns.
I generally, when hunting, carry a small container (1 or 2 oz) of the windshield washer fluid, and it alone is sufficient to clean my rifle spotless. Sometimes I will put in a bit of Murphy's (and honestly cannot tell you why...), but see no difference in the end. The methyl alcohol is a wonderful drying agent, so little or no moisture is present in the barrel when it has been used and followed with a dry patch.
Your recipe is different from the ones I have seen, using equal parts. If I recall accurately (which, at my age can be either a challenge or an oxymoron), the formula presented to me was two parts wiper fluid, one part of Murphy's, one part H2O2. Too sudsy for me, and, as I said I found the ww fluid itself to work well. I keep a spray bottle in my range box, as opposed to the little bottles that go hunting with me, so we can clean up at the range and not have to mess with it at home (I be lazy).
So, that's my understanding of the concoction. Mostly anecdotal, but I have shot MLs for a week or two and have seen success with the wwfluid, but others have used other formulae that worked well for them. Your mileage, however, may vary.
Cheers!