I'm in the process of doing this now, using Postmark's SMTP relay. Hopefully one day I'll meet their requirements for not retaining logs:

1. Account must be Approved and in good standing for at least 6 months

2. Account must be on a paid monthly plan for 6 consecutive months (or have purchase credits)

3. Account must have a lifetime sending volume of over 10,000 messages

https://postmarkapp.com/smtp-service

https://postmarkapp.com/support/article/1139-can-i-hide-or-d...