Hmm, how are you deploying things that a database costs $30/m? I just put it on the same server as the application worker and split it off if I need more performance later.
Since it's free below 10K rows and only $9 for 10M. And, the dataclips feature always comes in handy.
For backing up postgres, all you have to do is setup a cron job that backs up the postgres' data directory to S3/Google Drive/Dropbox every hour/day.
If you want proper replication and failover then you can probably use 2 digital ocean droplets each for $5/month and another $5 VM for the application server itself.