I have an idea to isolate the battery supply with a diode so as not to charge the dry cells when mains power is on.
Question: will a diode in series with the dry cell supply impact on noise levels at the cell output?
Many (if not most) of these have been modified for mains operation. All you need is a two diode OR circuit as shown in the picture. The external supply voltage should be higher than a fully charged battery. There is plenty of battery voltage, no need to worry about the diode drop. Check with an oscillosope if the original electrolytic capacitor needs to be replaced.
In case of a switching supply an additional linear regulator stage (e.g. LM317) would be a good idea.
Always disconnect the external supply (cable!) before wiring your test setup. The leakage of some poor power supplies can be high enough to damage your equipment. At least it ruins the measurement.