Thank you all for your replies. I've been following the suggestions and trying some things....without success, until today. Yesterday I completely removed (again) all SMTP settings and recipients, and added them back in. This time however, in the return address for the BE sender, I entered in a live email address (rather than just noreply-backup@). So this morning I still did not recieve the job notification email, however there was an email in the sender return address mailbox. It was an "undeliverable" notification stating "This message is larger than the current system limit".
So now that I know BE is at least sending the notification, I went back into Options, and checked the job log settings, it was set to "Summary information, directories and files processed". That would explain the huge job log.
So I have set it to "Summary information only", and will cross my fingers for tonight.
Cheers. Shane.