Seeing this in the related questions list: SoundCloud, download or stream file via api
led me to further examination of the API docs ( http://developers.soundcloud.com/docs/api/reference#tracks ), where I found that the track properties do include download_url and duration.
So the answer to my question is "yes, it is possible".
Edit as of September 2, 2013:
I was able to make a download link, but only one that initiated a download dialog, and not a URL for an mp3 file that would be appropriate for me feed's enclosure tag.
I tried e-mailing the SoundCloud api support e-mail address, but got no response. I then tried their general support e-mail, and did receive a reply telling me that the answer to my question is NO.
SoundCloud's api does NOT support .mp3 URLs to drop into the enclosure tags in my pre-existing feeds. It was instead suggested that I apply for their podcasting beta, which I will now investigate.