As SAM Broadcaster's statistic relay function only works with standard http streaming configurations of Icecast and SHOUTcast, unfortunately there is little we can do to enable it.
We do provide a public facing API which contains the current connected listeners etc,
This is contradicting.
You provide a public API for the connected listeners.
I understand that you cannot expose a Shoutcast/Icecast server to the public (probably because you don't use any of them for your infrastructure).
However for SAM (and other software compatible with those stream servers) to read the statistics, you don't need a Shoutcast / Icecast server, just like you don't need a Shoutcast/Icecast server to accept stream sources in that format.
The statistics part is especially easy if you already provide the data in any format.
You just need to put together an XML page containing the data in a certain format. (IIRC for shoutcast you have the public 7.xml/7.html and the admin.xml for more detailed statistics)
The XML formats of Shoutcast in version 2 are thoroughly documented. No idea about the formats of Shoutcast v1, but that format could be considered well known and either should be easy to provide.
I.e. if your API currently returns JSON or even an HTML page at radio.co/api/streamstats/<userid>/stats.json or stats.html (both completely made up, I've got no idea about how your API looks like), just add a radio.co/api/streamstats/<userid>/7.xml radio.co/api/streamstats/<userid>/7.html and radio.co/api/streamstats/<userid>/admin.xml with the proper formatting to make it look like it came from shoutcast/icecast.