I don't think that a filtered power supply will change anything. The filtering will be effective for low frequencies, where the DSO has plenty filtering itself, whereas the power cord will still act as an antenna for high frequency radiated signals. The latter are much more likely to enter the DSO via the DUT and probes anyway.
A good test can be set up with the DSO itself:
Look at the screenshot below and copy the settings:
Inputs: left open (nothing connected)
Trigger: mains
Vertical gain: 500µV/div, 20MHz bandwidth limit
Timebase: 50ms/div
Math: FFT, 10 times averaging, Flattop window, 1Mpts, reference level -60dBV, Scale 10dB/div - as can be seen in the screenshot.
[attachimg=1 width=800]
SDS1104X-E_Noise_50ms_500uV
The search conditions for the peak table were set to -120dBV (1µV) threshold and 10dB Excursion.
As can be seen, the strongest spur up to 1MHz is not near the mains frequency, but at 1MHz at -107dBV, which is less than 5µV RMS.
If your test shows no spurs stronger than, say, -80dBV in that frequency range, then the internal noise (including the one coming from the power supply) would certainly not be significant for your measurements.