How can I reduce an oscilloscope's bandwidth to that of an Arduino's ADC?
To reduce the bandwidth of an oscilloscope to "Arduino ADC bandwidth", you need to know all properties of the Arduino's analog front end - buffer amplifier (if any), bandwidth/dynamics of its sample-and-hold unit, etc. Which you don't know. And stability of reference voltage and power rails of the Arduino board will reap its toll as well.

Therefore an attempt predict the internal noise level by reducing the sampling rate of an (external!) oscilloscope to 10ks/s will reveal absolutely nothing about what gets sampled inside the Arduino. The memory depth has absolutely nothing to do with this situation, and the sensitivity of scope has nothing to do as well. So take the advice from Chris Stratton and research statistics of what the Arduino ADC is producing, using various input signals, zero short, zero with some reasonable impedance, sine wave, etc.