How to implement an RF Mixer for chirp signal in python?
Wouldn't mixing them give you a sum and difference frequency and you have to lowpass filter to get the difference frequency only?  Also your chirps are the same frequency, just out of phase?  So the difference would be zero and the sum would be double.
So:

Actually create a chirp that is a different frequency from the original, not just out of phase.
After mixing, lowpass filter to keep only the difference frequency