OK, it turns out that I just need to add "&bucket=audio_summary" to my Song/Identify query, and I get everything I wanted in a single query. And adding more buckets gets me more info. Sweet.
Just in case someone else has a similar question, my query was: http://developer.echonest.com/api/v4/song/identify?api_key=API_KEY&bucket=audio_summary&bucket=song_type