As I said , the database for Solo to use IS set up in System Configuration, and it does work for all of the other profiles in this Solo installation.
I'm assuming that the database used for Akeeba Solo is dedicated for that use, and that connection information goes in System Configuration. And that inside each profile, the database connection information is for a database with the site being backed up. Are those assumptions correct? If so, then we have it set up correctly.
So I'm not sure exactly what you are telling me. Are you telling me to put the Akeeba database info into the profile that isn't working? Won't that attempt to back up the Akeeba database with the site?