Hi Kerry,
According to your post, if you have configured your Gmail account in Outlook, I suggest you use .pst files to transfer your emails.
Please refer to the following steps in the Outlook client.
1. Click FILE> Open & Export> Import/Export.
2. Click Export to a file> Outlook Data File(.pst).
3. Choose the folders which you want to transfer.
4. Save the .pst file.
5. Click Import from another program or file to transfer the .pst file to your Office 365 account.
Here is an article for your reference.
http://support.microsoft.com/kb/287070
Using .pst files could transfer all emails including archives.
Since Gmail is different from Office 365, the fairly complex label hierarchy you mentioned may not be saved(the folder structure can be saved) after you transfer the emails.
Could you provide more detailed information about the fairly complex label hierarchy? Maybe we could provide you with you a workaround.
Thanks for your understanding.
Best regards,
Kid Yang