I've found the solution and am documenting it for anyone else with this specific problem.
When Joomla returned the error from the MySQL Database, it attempted to find the file by reading it. Running 'chown -R mysql:mysql /var/lib/mysql' set all the permissions back to normal, allowing the database to be read.
In my attempts to repair the databases, I neglected to check ownership of the files. (As I was logged in through SSH as root, yes, shame on me.)