Keep in mind that the binaural beats community spouts a lot of BS.

Only for the lowest frequencies (say, under 100 Hz) are there reliable traveling waves to even measure in the brain. Due to the firing refactory period where a neuron resets its membrane potential, most neurons have a max firing rate in the range of 250-300 Hz, so it's not possible for difference tones to explain any binaural beats above those frequencies.

Even for lower frequencies in the range of delta-gamma waves, it seems deeply unlikely that difference tones are the explanation, because the neurons encoding for frequency are separated spatially, so it's not as if two traveling waves reach the same neuron, subtract out, and result in the difference frequency.

However the brain produces the beat frequency, it's nothing like the way signals subtract in other media.

Any nonlinearities in the Fourier transform process (of which there are presumably many) will manifest as harmonics of the beat frequency. I'm on my phone right now, but if you take two simple-ratio tones, add them, apply a nonlinearity (like exponentiation to a non-1 power), and then Fourier transform, you will see the beat frequency and its effects. I imagine (but do not know) that this is the physiological mechanism for high-frequency beat frequencies.

For binaural beats with low-frequency tones, I can only guess, but you can actually reconstruct phase information from the frequency domain signal depending on your sampling period. As a trivial example, imagine f << 1/T; now you can treat the DC component of the Fourier transform as a time-domain signal that contains f. I imagine that's how binaural beats work, as they occur at sufficiently low frequencies for this to happen. It could also be that the ear transmits low-frequency time domain information as well as frequency-domain information.