What's the best way to play multiple characters each with different mods? I looked into the Mod Organizer, which seemed to be the solution, but it doesn't seem like it will support using stuff like BOSS to sort the load orders on each list. What is everyone using to pull this off?

Separate installs?

I know it was easy in Morrowind, once the game was installed, the game directory could be copied to a new location, then use a short cut to that exe. I have never tried this in Oblivion, but it would be quick to copy the root game directory, make a shortcut to that Oblivion and see if it works. The only issue is, it would use the same directory for your saves, so that could be a problem, unless you name them right.

This might work for the saves issue

http://www.gamesas.com/topic/1414861-save-profiles/

The only other way would be to have two separate Data Files structure for each character/mod mix, then move those folders in an out as you play each one.

Setting up seperate profiles with Mod Organizer is the best way to do this.
You can use utilities like Boss and TESVEdit with MO. http://wiki.step-project.com/Guide:Mod_Organizer#tab=Third_party_programs