How to deal with overlap in STFT when considering the highest energy among multiple channels
The obvious approach to this would be a crossfade between channels. This leads to problems, just as obvious, as you would get nasty comb filter effects at 2000Hz and above. So this is a no-go.

A solution just as simple would be to use just one channel, but apply a gain at each window, so that the energy is aligned with the one of the loudest channel at that point in time, i.e. the gain would be 
$$a_i = \frac{\max\limits_{c}(E_i^c)}{E_i^1}$$
for $c$ as channel index, $i$ as time index and assuming you use channel $1$.
This way, you avoid any phase problems whatsoever and still have a signal, that suffices your energy requirement. Also, there is no non-linear processing and thus, no artifacts resulting from it, which would most certainly occur, if you follow some approach with time aligning the signals from different channels to avoid comb filter effects.

EDIT: I know that the overall SNR will not be improved by this approach, but I think the advantages are worth it, since SNR differences should be quite small, as distance between microphones is also small.