Battery power to DC mains power
It could be that the initial surge of turning on the alarm is causing the voltage to drop too low.  Many electronics devices contain "brownout detector" logic that resets the system if the voltage drops below some threshold.  Alkaline cells have a pretty good pulse current capability, perhaps better than your external supply.

Energizer has a good overview of battery internal resistance on their website.  Internal resistance is measured by measuring the voltage change due to a step change in current a R = dV/dI, as shown below. 



Energizer says that the typical internal resistance of a new cell is about 150 to 300 milliohms.  At 3.5A, the current rating of your supply, 4 new cells would drop the voltage only about 2V, assuming 0.15 ohms internal resistance.  That would still leave the voltage at the low end of the operable range for an alkaline cell.

You can try putting a large electrolytic capacitor in parallel with the battery terminals.  That could perhaps smooth out the initial surge and allow the external supply to work.

You could also take the device apart some more to determine what it is using for the siren and look up the peak current requirements, to see if they are within the capabilities of your supply.

If your meter has a min/max capability, you could try measuring the minimum voltage in the battery and external supply cases and see if the external supply is dropping lower when the siren turns on.

One final thought, 220ma in standby mode seems very high for any modern low power electronics device.  Perhaps the alarm is defective.