I got this to work using wget instead of curl, but I'd still be interested in knowing how to make this work with curl if anyone has pointers.
This command works:
wget --delete-after --auth-no-challenge --http-user=${userID} --http-password=${authToken} http://server:8080/job/job_name/buildWithParameters?token=runme\&ENV='dev-1'\&PLATFORM='desktop'\&PRODUCT='Test'