Synchronizing Two Audio Tracks
Chroma-based approaches are used when the tracks to align are two performances of the same material with widely different instrumentations - in which case you need to find the "lowest common denominator" which is harmony, melody, chord progression etc. Chroma are good for that - they totally abstract timbre, recording techniques. But you don't need those in your case.

As you have guessed, cross-correlation of the raw audio would be 1/ not very robust to noise, 2/ not very robust to differences in transducers (the two microphones might have very different responses) 3/ very costly if you have minutes of audio.

I suggest you to:


Extract a sequence of MFCC vectors from both signals. This way you'll have something of lower dimensionality and a bit more robust to noise and differences in transducers.
Optionally normalize the MFCC ("by design" coefficient 0 has more variance than coefficient 1 and so on...)... Ideally you'd want each row of the MFCC matrix to have variance 1.
Compute the cross-correlation of two matrices you get, along the time axis (that is to say, compute the 2D cross-correlation and just keep the data for the time axis).