You can use BASS_Mixer_StreamAddChannelEx to delay the start of a mixer source. For example, if you have a delay in milliseconds, you could do something like this:
BASS_Mixer_StreamAddChannelEx(mixer, source, flags, BASS_ChannelSeconds2Bytes(mixer, delayms / 1000.0), 0);
Please see the BASS_Mixer_StreamAddChannelEx documentation for details.