The DC resistance in parallel with the capacitor must be measured as it affects the AC measurement. If the diode conducts during either measurement, the results are not accurate.
This is what the B&K 881 and EDS-88A II do that. Nothing else seems to take into account. One of them uses a 25 mV signal for testing to avoid turning on a diode junction.
An inductor in parallel with the cap makes it rather messy. Fortunately, those don't appear in power supplies that I know of. They are usually series tuned to short the undesired frequencies to ground.