We found a workaround for the problem. I can use the API to get a build ID based on a build name, and in this instance supplying the <config id>.latest_successful works. So assuming I knew the name of the configuration I wanted to use, I could do something like:
http://dpbuildfarm/rest/ids?configuration_path=<configuration path> to get the config ID.
http://dpbuildfarm/rest/ids?build_name=<config ID>.latest_successful to get the last successful build ID.
http://dpbuildfarm/download/<build ID>/artifacts/** to get a zip of all the atifacts
Hopefully this helps debug the problem?