Hi ITAdmin1380,
I understand that you encounter the security alert issue when manually connecting Outlook to Office 365 after performing the staged migration.
I would like to confirm if you’re using the staged migration steps provided in: http://technet.microsoft.com/en-us/library/jj874018(v=exchg.150).aspx? If so, after a migration batch has finished running and you’ve verified that all mailboxes in the batch are successfully migrated and the initial synchronization of mailbox items to Exchange Online is complete, it’s recommended that you convert the on-premises mailboxes in the migration batch to mail-enabled users. Why? After a staged Exchange migration, a user has an on-premises mailbox and an Exchange Online mailbox. Because mail sent to the user’s on-premises mailbox is forwarded to their Exchange Online mailbox after migration, users need to connect to their Exchange Online mailboxes to access their email. But if a person uses Outlook to open their mailbox, the Autodiscover service still tries to connect to the on-premises mailbox. After you convert on-premises mailboxes to mail-enabled users, the Autodiscover service uses a mail-enabled user to connect Outlook to the Exchange Online mailbox after the user creates a new Outlook profile.
I noticed that you want to back up the mail items at this time and you haven’t converted the on-premises mailbox to mail-enabled user yet, the issue may occur because the Autodiscover service still tries to connect to the on-premises mailbox. Therefore, to solve the issue, you will need to bypass the on-premises Autodiscover checking during the backup (You can convert the mailbox to mail-enabled user after the backup is completed). However, consider that the bypass setting is related to the on-premises server part, which our forum doesn’t focus on it currently, to get dedicated support, it’s recommended that you post the issue with details in our on-premises Exchange server forum to check if they can provide some information for you. Thanks for your understanding.