Will noise cancellation with playstation eye improve my spectral analysis?
As I read your prompt, it seems that the real question you’re asking is why you can’t record frequency peaks greater than 7kHz. The second question seems to be whether you can improve this performance through noise cancellation techniques.

If that’s the case, and if you’re right about the 2nd and 3rd channels being reversed, then I think your summation may introduce frequency filtering effects in your output and not necessarily true noise cancellation. This can either work for or against you. But this would depend on the orientation between the microphone and speakers, since in your link it appears that can affect the delays between the channels. Either way, I’d go with the single channel to simplify things.

And if you’ve only been recording music and not individual tones at greater than 7kHz, then it also makes sense that you’re not grabbing high frequency peaks since audio energy tends to be concentrated at lower frequencies. There’s a reason why telephones and AM radio cut off at around 5kHz. You might also be attenuated because of sinc roll off during the sampling process, although I suspect that’s not the big issue here.

Have you tried recording the entire spectrum, instead of just the peaks? And have you tried comparing that against the FFTs of the original audio files?