OutbackV2008,
All correct. It's pointless for me to explain a bunch of steps, the whys and the logic, when we already know the answer.
The source electronics must come off the factory harness and share a respective (separate + & -), common point with ALL other audio components in the audio path. You can use an interrupt relay off the key or switch so that you can maintain the keyed or switched function but circumvent the key or switch power.
It's normal for the tower HLCD speakers to have a greater noise level because they are 10 times more sensitive in acoustic terms.