I haven’t ever seen the code that runs the clock,though I’ve considered getting one at some point. Are you decent with programming in general? I ask because I personally don’t find programming an arduino to be much different than programming anything else. Especially if you do c or c++ sometimes, you’ll be right at home.
Anyway, off the cuff I wonder whether you could take the ground connection — or depending on the circuit, the positive lead, whichever is shared — from the LEDs and hook it into an arduino pin. Setting that pin high or low would then make it possible or impossible the diodes to light. Might be able to do this in conjunction with some simple checks of the time to easily flip your display on and off.