I find that if you write to /devices/DeviceID/output, it'll work if you set the Accept header to "application/vnd.littlebits.v2+json", the Authorization token as you'd expect, the Content-Type header to "application/json", and then include the following in the payload:
{ 'percent': 100, 'duration_ms': -1 }
The docs seem to be out of date. I found some examples online when I was writing the code to do this, it seems the APIs have gone through several revisions but it's not clear.