The ARC1319I message indicates an enqueuing issue -- that is, your backup job wants exclusive access to the data set and another job (or CICS region or started task) already has the data set allocated. I'm not sure why you think this problem would be related to the Auto Backup setting, but there will be little (if any) correlation between the two. The manual DFSMS Implementing System-Managed Storage
has this to say about the Auto Backup setting for Management Class:
Y for Auto Backup to ensure that the data set is retained on both DFSMShsmbackup volumes and migration level 2
And how does this relate to whether or not a data set can be enqueued for backup?