I heard back from Microsoft that...
- Currently, the On-Premises Gateway does not support OAuth (a requirement for SharePoint Online connectivity); so if you have a dataset that has both SharePoint Online and on-premise data sources, that dataset cannot be refreshed using an On-Premises Gateway.
- As mentioned here, you can use a Personal Gateway to set up a scheduled refresh of a dataset that has both SharePoint Online and on-premise data sources. This is what they recommended.
- Microsoft is investigating the possibility of a change to allow OAuth with the On-Premises Gateway, but have have no timelines for it.
- They recommended that I continue to to use the On-Premises Gateway for datasets where it works, but use the Personal Gateway as a "workaround" for my one dataset that has both SharePoint Online and on-premise data sources