Another way you could implement a delayed fade-out is with an envelope, eg. the BASS_FX add-on's BASS_FX_BFX_VOLUME_ENV effect. It could look something like this:
lVolFxOut = BASS_ChannelSetFX(AudioStream, BASS_FX_BFX_VOLUME_ENV, 0);
BASS_BFX_VOLUME_ENV param;
BASS_BFX_ENV_NODE nodes[3];
param.lChannel = -1;
param.lNodeCount = 3;
param.pNodes = nodes;
param.bFollow = false;
nodes[0].pos = 0;
nodes[0].val = 1;
nodes[1].pos = aDelay;
nodes[1].val = 1;
nodes[2].pos = aDelay + aSec;
nodes[2].val = 0;
BASS_FXSetParameters(lVolFxOut, param);
Note that the BASS_FX add-on needs to be loaded before you use any effects from it. You can force it to be linked/loaded by calling a function from it, eg. BASS_FX_GetVersion during initialization.
If you're using BASSmix to play the stream then another option is to use BASS_Mixer_ChannelSetEnvelope instead.