Changed
Request
Need a quick way to pick a favicon per multisite.
Discovery
The theme has a hook for setting the path for responsive favicons.
Each site needs to override all of those paths, so most options are limited if they only do favicon.ico.
Implementation
Try these modules:
Metatag Favicons (Metatag submodule)
Responsive Favicons (Looks like this could work)
Environment Indicator (Different icon per live/dev/local).
If those don't work, add settings to the theme to configure overrides for those paths. May need to provide a upload form for a zip or individual files.