Very difficult to solve noise problem

OK, this will be a long-ish explanation of EXACTLY the symptom I am experiencing and the troubleshooting steps I have already gone through.

Bottom line: A pretty load pop from speakers when turning key to "start"

Car:2002 BMW M3
Head Unit: Sony MEX-GS810BH
Amplifier: JL Audio 1000/5v2

I have the pop ONLY with this EXACT chain of events:
Car sitting. Get in car. Turn key to ACC or run and allow radio and amp to turn on. The moment the key turned to the start position, the radio display goes black and I get the pop.

If I have just driven the car and attempt to reproduce it, on the first restart of the car, radio display WILL NOT turn off and there will be no pop.
If I shut it down and do a second re-start of car then the pop comes back.

ALSO, if car is connected to a battery tender I can start the car over and over again without the display going off and without getting the pop

I don't particularly care that the display of the radio shuts off momentarily, but I am trying to eliminate the pop.

There is no other audible noise in the system.

Again, the pop is not occurring when the radio lights back up, it is happening the moment the starter is engaged.

Here is what I have tried:
Connecting the remote turn on lead to amp with radio already powered.
Amp turns on, no pop
Removing remote turn on lead from amp while leaving radio powered.
Amp turns off, no pop
feed remote turn on from constant +12V so amp stays on for test and turn radio on. No pop. Then turn radio off (while amp remains on). No pop. Turn radio back on (still no pop) then turn key to start. POP (same as when remote turn on is wired correctly.

I then tried to eliminate the +12v remote turn on completely from the equation: I disconnected the remote turn on connection and moved the turn on switch on the amp to signal sensing. Same results except instead of one loud POP, it became two quieter (but still very much there) pops.

Pop is completely absent if RCA cables are not connected to amplifier. Pop does not vary based on where volume level on head unit is set.

Since the head unit has 5V pre-outs, I have tested with the input level switch on the amp set to "low" and set to "high". Pop is there either way.

Please, some ideas as to what is happening?
And more importantly... Ideas on a solution.

-pete