Is it possible to connect Power BI to Anaplan such that I could import Power BI entities or tables into Anaplan?
Sort of, but not persistently.
To import data into Anaplan from ANY source you will need to utilize the Anaplan APIs. In some cases there are third party tools that have built in connectors, like Informatica, but it still relies on the API set. To import data into Anaplan you will need to write a script in Power BI than can generate a CSV or Text formatted file that can then be called via a RESTful API. My preferred method with Power BI is Python but there are other ways. Read @chase.hippen excellent article on python and Anaplan. Somebody wrote a good article on Power BI somewhere. I'll look around for it.
If you can live without an on-demand connection to Anaplan, like with Direct Query (power BI) I would recommend you utilize a data mart or data lake for all your reporting and visualizations. The data can be centrally managed rather than out of someones Power BI report. But that's a whole different conversation.
Hope this helps. Keep the conversation going. Let's get you to the finish line.
Jared, JWakefield, thanks both for your feedback. The Python route sounds very promising, that may be the way we go. Thanks!