It sounds like you want to play a clone of the mixer? You could use a DSP function and "push" stream to do that, ie. the DSP function feeds the mixer output to the push stream. Something like this:
HSTREAM clone = BASS_StreamCreate(freq, chans, flags, STREAMPROC_PUSH, 0); // create push stream with same format as the mixer
BASS_ChannelSetDSP(mixer, CloneDSP, 0, (void*)clone, -1); // set cloning function on mixer
BASS_ChannelPlay(clone, 0); // start the clone
...
void CALLBACK CloneDSP(HDSP handle, DWORD channel, void *buffer, DWORD length, void *user)
{
HSTREAM clone = (HSTREAM)user;
BASS_StreamPutData(clone, buffer, length); // pass the data to the clone
}
Please see the documentation for details on the mentioned functions.