You need a Static IP for your WAN(Internet) connection.
Your ISP needs to allow outbound communication over port 25.
You need to add the line ip4:Your IP to your SPF text record in DNS.
Then log into your Office 365 global admin.
Go To > Exchange Administrator > Mail Flow > Connectors.
Create a new connector
Select from your org to O365
Validate by IP.
Enter your IP in the given field.
In your copier you will use your mx record for the SMTP server address, select port 25 (or 587) and disable SMTP Authentication.
Now as long as your from address is a VALID email on your o365 hosted domain it will allow you to send email to addresses within your domain without SMTP authentication.