Issue
I'm unable to use dataclips with the database because it doesn't show up in the dataclips dropdown "Choose a database".
Resolution
If you just just created the database there is a chance that the database is not yet synced with Dataclips system yet. Please try visiting https://dataclips.heroku.com to trigger a sync process, and wait for 3-5 minutes, and then try to create a dataclip again.
Alternatively, you will want to make sure that you have Operate or Deploy permissions on the app. View permissions isn't enough to allow you to create a dataclips for a database.