Frequency dependent phase difference between signals arriving at two microphones
For a constant time difference, the unwrapped phase difference measured at the centers of the two windows is linearly related to frequency. (To measure phase at the center of an FFT window, use an fftshift prior to the FFT, or flip the phase of the odd numbered FFT result bins.)  So you can use all of the FFT results, not just one bin.   The low frequency bins may help in determining the potentially ambiguous phase unwrapping of the higher frequency bins, if the signal is sufficiently wideband.