In case anyone else is looking to do this, I solved the issue using the Recent Posts Widget Extended plugin. Using a shortcode to pull from a specified category and limiting to 1 entry. This pulls the title and excerpt from the most recent podcast episode.
I'm also using the playlist player with a custom style that hides the title and playlist. (Also limited to 1 entry.)
The result is a podcast homepage that features the Title, Description and Player for the most recent episode. Without having to manually update the page every time.