SMS Backup and Restore saves the backup as XML if I remember right, so you could open that in a text editor and check whether the outgoing messages are in there (may be more comfortable on a computer). That would narrow the problem down, i.e. tell us whether it's the backup or the restore that isn't getting the messages.
But before going to such efforts, are these just regular SMS? I found one discussion elsewhere where other Samsung users were having problems backing up outgoing messages and someone suggested that a feature called "advanced messaging" is the problem, which I think is a marketing name for RCS. Certainly SMS Backup and Restore can't handle such messages, so that sounds plausible, though you'd expect Samsung's own tools should be able to (but never rule out the power of oversight or lack of communication between software teams). So I'd check whether you are using a feature called Advanced Messaging, because if you are that might be the cause.