Well, I'm not quite sure what happened. I tried unhiding the BE user account, but it didn't work immediately. I also tried it a few hours later, just in case there was a propagation issue.
I ended up creating a selection list for just the mailboxes, selected the server, selected resource credentials and selected the existing BE user account. That worked for that job.
I did the same process for my daily/weekly/monthly selection list and now it seems to be working again.
I'm not quite sure what really happened to get it working again.
I'm going to hide the BE user account again because I don't like to have service accounts showing in the Global Address List. That confuses users and they sometimes accidentally select the wrong email account.
If it fails, I'll unhide it.
Either way, I'll post the answer I find tomorrow evening.