Aftermarket Fuel System Electrical Noise

I believe a few of you have expereinced issues with long cranking cycles when installing Fuel systems upgrades that remove the factory fuel pressure regulator. That is because the factory signal to the pump is not completed until the key is in the "Run" position. The pump then has to fill an empty circuit which takes a few seconds until it hits the rails. The harness that bypasses the OEM ground switching to full time grounds solves this issue, but i noticed there is prominent hum in the speakers from the fuel pump. If i take the bypass harness out this noise goes away. So i am looking for a solution that gives the fuel pump priming effect by turning the pump on with the Key in the "On" position, but no radio noise. And it needs to be something simple. That was the whole point of this harness to begin with.

I am thinking a relay to switch the grounds, but cannot wrap my head around how to wire it efficiently. Another thought was if there is some type of ground noise isolator that can be wired in.

Look forward to hearing your thoughts.

The 2 images on the left are factory wiring and the image on the right is with the bypass harness.