When MIBRO and I realized our mistake in purchasing the wrong power supply, he returned the first one and bought this one. MIBRO also bought these splitters to simplify tapping into the power supply to activate the sound relay.

Above, timer and power supply. Below, plugged into SKELLY. 

MIBRO then plugged this new power supply into his sophisticated timer, put the dummy batteries into SKELLY's back, left SKELLY's power switch in the ON position, and then let the timer activate.

Nothing. Apparently silently turning on the power did not result in "dancing". 

If the DEMO button was pressed immediately after being powered, SKELLY would dance for the full 30 seconds. However, permanently shorting the DEMO button and leaving the switch in the ON position did NOT result in SKELLY dancing when the power was turned on. Apparently, the DEMO button had to be pressed just after powering on.

Now the parameters of the control electronics were clear. Just after being supplied power, a relay would close to substitute for the DEMO button on SKELLY. At the same time, another relay would close to "press the button" on the sound module.