I leave the only way it has worked for me in two Multisite installations in case it is useful to anyone:
– Uninstall the Font Awesome plugin
– Install the Font Awesome Cleanup plugin and activate it on all sites https://github.com/FortAwesome/wordpress-fontawesome/releases/tag/4.3.0
– Install again the Font Awesome plugin. It will fail…
– Launch the cleanup on all sites.
– Activate the Font Awesome plugin for all sites. It works!
