With an amplitude modulated signal, why does a bandpass filter not filter out the signal of interest?
In the frequency domain, AM modulation ("multiplication") is the convolution of the modulated signal with the carrier.  Thus, the Fourier Transform energy is at the carrier frequency, and has the bandwidth of the original signal.

So, as long as the width of your lock-in filter is as wide as the bandwidth of the original signal, you don't lose signal.