You can use BASS_ChannelLock to lock the mixer before un-pausing all of the sources to ensure that they resume together in the mixer output. For example:
BASS_ChannelLock(mixer, true); // lock mixer
BASS_Mixer_ChannelFlags(source1, 0, BASS_MIXER_CHAN_PAUSE); // unpause a source
BASS_Mixer_ChannelFlags(source2, 0, BASS_MIXER_CHAN_PAUSE); // unpause another source
BASS_ChannelLock(mixer, false); // unlock mixer