Here's how I've gotten Code Extinguisher to work when having the app outside the root of the site.
1. I’ve moved the entire codex app folder to my general, multi-site CI applications directory, which is /ci/apps --outside of the public root of my site -- so, in my case, the relative url from my site root to my CI apps directory is “../ci/apps/”. (When I say entire codex folder, I mean I placed the codex folder which itself includes the "application" folder and the "assets" folder in my /ci/apps folder.)
2. I've placed backend.php in the public root of my site, and then I’ve renamed the “backend.php” to “form.php” (because I'll use it to create public forms for the site and I want a meaningful url).
3. I’ve changed the apps index page setting from "backend.php" to “form.php” in /codex/application/config/config.php.
4. I copied the assets folder from the codex directory into the root of my site. I named it "ce_assets".
5. In /codex/application/config/codex.php, I’ve modified the following line of that file as follows:
You've told me that having multiple assets folders is unwise, as it interferes with updating the application. So, I've deleted the content in the default assets folder, under the the codex directory. I placed a text file in that folder telling to where the assets folder content has been moved.