Lock-in amplifier output to spectrum analyzer
The demodulator can be very accurate. It's not just a diode- it may be a precision analog multiplier or a digital DSP function. I've gotten ppm accuracy using digital methods. 
Yes, the spectrum analyzer will only see frequencies from approaching DC to the cut-off frequency of the low-pass filter. Of course the filter will not be infinitely sharp so there will be some attenuation of frequencies approaching the cutoff frequency and some higher frequencies will get through.