How to improve SNR by averaging multiple identical sensors - hardware or software?
When you average AC signals, the answer is zero, unless EVERYTHING is phase-locked.  This means every one of your microphones must be equidistant from the sound source, or errors will be introduced, dependent on the difference in distance and the frequency of the sound.

For 16kHz, the wavelength is 2.1cm, so this is a real problem.

For this reason, I would NOT average the raw mic data.  I would process the data, demodulating as you mention, and then average the results.

So, I would lean toward the SSH solution, process each channel, and then average the resulting outputs.  

Alternatively, you could demodulate then sample, and then use either technique.