Complex signal low pass filter
You're not actually mixing two complex exponentials, because otherwise you'd get a complex exponential with the sum of the frequencies of the individual complex exponentials. That was already mentioned in a comment by Marcus Müller.
What you are doing is compute a signal with real part $\cos(\omega_0t)\cos(\omega_1t)$ and with imaginary part $\sin(\omega_0t)\sin(\omega_1t)$. If you lowpass filter that signal, only the components with the difference of the two frequencies remain, but these components are the same for the real and for the imaginary part, because
$$\cos(\omega_0t)\cos(\omega_1t)=\frac12\big[\cos[(\omega_0-\omega_1)t]+\cos[(\omega_0+\omega_1)t]\big]\tag{1}$$
and
$$\sin(\omega_0t)\sin(\omega_1t)=\frac12\big[\cos[(\omega_0-\omega_1)t]-\cos[(\omega_0+\omega_1)t]\big]\tag{2}$$
Clearly, the components with the difference frequency are the same.
Also, if you want to filter out a component with frequency $\omega_x$ (in your case the component with the sum frequency), it is not wise to use a filter with cut-off frequency $\omega_c=\omega_x$ because the attenuation at $\omega_c$ is usually quite modest, in your case $3$dB. Choose a frequency that is much closer to the lower frequency component to get a decent attenuation of the higher frequency component.