Can my django app act as a header-stripping proxy?
I don't think it's possible. It should already be entirely up to the browser. For example Opera asks user weather to open it or download.
You could embed them in your site using their Embeded Code.
EDIT:
Nope you can't even use the link provided by the header since they've worked around hotlinking. Each download is associated to the browser/session so you can't store the real url for the mp3 and link to that.
Public Soundcloud track urls force a download upon browsing to them by utilizing a Content-Disposition header (I think.. ) which triggers a download for a known mime-type. Is there a way to proxy (create a passthrough for) these urls and strip this header from my request.  I want to avoid serving the mp3 myself but I don't want to trigger a download.