Using Outlook 2016 (Office 365)
Create [Gmail] IMAP account
Subscribe only to following Required Folders:
You can then MOVE or COPY the [Gmail]/Sent emails to local POP archive.pst
It is not possible to create a rule that would move or copy the messages automatically due to Outlook 2016 requiring a "only on this computer" clause to the rule.
Spark iOS app install/and testing.