My research has led me to learn that Apple\'s media element handler is a singleton, meaning I can\'t have a video playing while an audio is playing in
This is currently not possible. As you notice when a video plays it takes up the full screen with quicktime and moves the browser to the background. The only solution at this time is to merge the audio and video together into an mp4 format and play that single item.
If I understand you correctly, you are not able to merge the audio and video together because it relies on flash? Since iOS can't play flash you should merge the audio and video together and use flash as a backup. There are numerous html5 players which use javascript to try and play the html5 video first then fallback to flash for backup.