Hello Nathan, if you do not wish to expose your DB's directly on a public IP, then we can recommend using AWS PrivateLink, we have documented this here:
https://www.gooddata.com/docs/cloud/connect-data/use-aws-privatelink/. Note that this is a paid service and/or available as an addon, but it may be possible to test it (subject to approval). If you are interested in this option, please let me know I will reach out to the proper parties internally on your behalf.