I was finally able to get this to work properly. The "Query contains unknown function name" error is slightly misleading. The actual issue is related to the personal gateway not being configured and linked to the PowerBI Service account correctly.
If you have the Personal Gateway downloaded and installed, make sure you sign in with your PowerBI account, verify that the Custom Connectors folder is correct and has correct permissions (if using the non-personal gateway). Verify the Network Ports test under the Diagnostics tab as well.
If the personal gateway is running correctly, you should see it appear in your PowerBI Service Data after you reopen the "Discover Data Sources" link under the error message.
I'm not sure why this setup was causing me issues yesterday, but it appears the solution is to re-install/tinker with the Personal Gateway settings in order to resolve the issue. To clarify, my custom connector is using OAuth authentication and the Web.Contents method to return JSON data from a REST API. When the gateway is linked to the dataset and functioning properly, you should get an OAuth credential prompt when attempting to set up the "Data Source Credentials".