I believe it's related to the keyless entry. You're hearing the same chime you hear when exterior lights are left on.
This happens when you are about 5ft from the car, hit the unlock button on your remote and immediately grab the handle. The lights flash twice and as they flash, your dash lights, center console light etc also flashes (which I think is dumb). Since the system is tied to the light chime, you open the door and catch the second light cycle and the blip from the chime.
Just for fun, unlock your door with the remote. Wait until the lights stop flashing (try this at night so you can see all the dash lights too). Open the drivers side door and hit the unlock button 1 or 2 more times to unlock the rest of the doors. Observe the chime. Now, turn your parking lights on with the door open and the chime should be a similar tune. I think there are 4 different tunes- key in the switch( door open), key to run position (6 beeps), exterior light warning and lastly the rapid beep when the car is moving and emergency brake is applied.
If this is not the case, there may be another issue. I believe that because the console/HVAC/radio backlighting also flash every time you use keyless entry, these constant and short periods of electrical load are the reason for the failure of the dash and radio light circuits.. My 2 cents.