For posterity, I popped in a copy of New Super Mario Bros. The system requested an update, and I did so, but little changed. So, I inserted a copy of Xenoblade Chronicles. Again, I was asked to update, and I did so.
Now, nothing changed when attempting to launch programs from the Homebrew Channel. MMM, Dop-Mii, etc. always launched using IOS60 through HBC, and never would it allow me to select an alternate IOS as it would crash every time. Using the LetterBomb exploit, I was able to get to MMM using IOS80. After the Xenoblade update, I was finally able to select IOS250 and patch IOS36.
Good grief, what a long, frustrating experience this has been. I've definitely learned to be more meticulous in my research when looking to correct an issue!
Thank you all for your guidance.