Just create a separate .pst file to use as an archive or backup or whatever and drag the emails (or copy them) from your Gmail inbox into that. Then you will have a local copy and can do what you like.
That's what I'd do in Outlook 2016. Not sure if you can create a local folder in Outlook 365. If you can't, then perhaps you could just use a local mail client to access the Gmail account (does Thunderbird still exist, etc?) and do it that way.
Alternatively, would
this solve your problem?