I definitely think lowpassing the output is the way to go. Since this frequency readback will be used for slow control of the beatnote frequency via auxillary laser temperature, even lowpassing at tens of Hz is fine. The jitter doesn't mean its useless, though.
If we lowpass at 16Hz, we're effectively averaging over 1024 samples, bringing, for example, a +-2kHz jitter of a 6kHz signal as you post down to 2kHz/sqrt(1024) ~ 60Hz, which is 1% of the carrier. This seems ok to me.