How to estimate the frequency characteristics of a nonlinear chirp?
How about just using $\dfrac{d\phi}{dt}$ to compute the instantaneous frequency?

GNU Octave code (I just made up a sample rate of 2 Msps):

pkg load signal;
x = csvread('nonlinear_freq_fit.csv');
y = hilbert(x);
z = y - mean(real(y));
dphi_dt = diff(unwrap(arg(z)));
Fs = 2e6;
f_inst = (Fs/2)/pi * dphi_dt;
k = [0:length(f_inst)-1];
t = k/Fs;
plot(k, f_inst);
axis([5000 5800 -2e5 2e5]);
title('Instantaneous Frequency');
xlabel('Sample index');
ylabel('Frequency (Hz)');
grid on;