Unwanted noise from bluetooth receiver

[spookybathtub]

Let me start with some background:
I have an 07 Hatchback, with factory radio including 3.5mm input.
I have a SoundGate PDCBLPAK hard-wired, which let me connect iPods using the dock connector for both power and audio. It's worked perfectly for 4 years, but now I have an iPhone 5 with lightning connector.

So I bought a Bluetooth A2DP receiver, the kind which gets power from a dock connector. Plug it into the SoundGate, and there's an awful noise. I think it sounds like a ground issue, but there is no problem when I plug an iPod in directly to the SoundGate. What could be causing this? The noise is partially drowned out when music is playing, but it still sucks.

I've tried grounding the SoundGate to two places: first a chassis bolt, then spliced
to ground wire directly behind the radio. Same problem both ways.

This is what it sounds like: https://dl.dropboxusercontent.com/u/...is%20noise.m4a
This noise was recorded with the engine off. With the engine running, there is another faint sound in addition, which corresponds to the RPMs.

Connection details on the SoundGate:
+12v spliced behind radio to ACC wire
GND spliced behind radio
audio line out spliced into the factory 3.5mm jack

I have now tried two different brands of Bluetooth receiver, with the same results.
Wireless Bluetooth Music Receiver
Mpow Bluetooth A2DP Receiver

I've tried pairing an iPhone and also a laptop, but it makes no difference.