OK, I made a discovery - the microphonic "ping" at power-up has nothing to do with the relay engagement. It is noise induced from flipping the power switch.
I tried leaving the power switch turned on, and instead turned the amp on and off by switching the AC to the adapter. The relay pulls in after 10 seconds. No "ping" noise (well, none in HD600s - I'm sure there is still some noise in efficient headphones or IEMs).
I also tried moving the power switch very slowly... also no ping sound.
It is true that if the delay is lengthened, you won't hear it as much when you flip the power switch. But I guess I don't quite understand why it matters. Let's say it takes 20 seconds (from the time you hit the power switch) for the ringing to calm down. You can make the power on delay more than 20 seconds and you won't hear it. Or you leave the power on delay at 10 seconds (which is where it is now) and you hear a decaying ringing for 10 seconds. It's not that loud...