Have you thought of using a USB soundcard rather than the 'scope approach - you'll need a calibration source to compare against(*) but you can simply save the signal in a WAV file and do all the processing off-line. Low pass filtering not needed.
(*) A simple metal-film resistor can be used if your low noise amp is JFET based. 6k at 25C is close to 10nV/√Hz
But the Tek MSO54 does the job very nicely. I concede that a soundcard is way cheaper. Whether soundcard or oscilloscope, limiting the bandwidth of the signal going into the digitiser is useful because it avoids you wasting ADC dynamic range on out-of-band noise. And, of course, reduces aliasing.
A USB soundcard already has filtering is the point, and it has sigma-delta ADCs so aliasing isn't an issue.