Send from Gmail through your own domain, and forward incoming Gmail into Axene.
Verify your domain and create an Axene SMTP credential.
In Gmail Settings, Accounts, add your domain address and choose to send through an SMTP server: mail.axene.io, port 587, STARTTLS.
To route incoming Gmail through Axene, add a Gmail filter that forwards to an address on your verified domain.
A native Gmail connector with OAuth, so you skip the SMTP setup, is on the roadmap.