Update on this: I had the car at the service center for several issues, this being one. They say they couldn't recreate the problem and to bring it by and show them the sound in person when I'm near the Goose Island SC. I have no plans to be down there for a while, so I'm going to try some things every so often when I get annoyed enough with it.
Yesterday I attached a GoPro to the front of the car to look at the shutters to see if they were making the sound, and to at least get it recorded. In the vid, I start off below where it makes the sound, stay up where it makes it for 10 secs, then slow down.
It didn't seem to be the passenger side shutters, but while I was down there I noticed the metal trim piece that goes around the lip at the bottom of the facia was kind of loose. So I grabbed some gaffers tape, taped it up, and took a test drive. No luck.
Next, I just started poking around different things in the front to see if something is loose. I noticed that the trim inside the driver's side headlight air channel was loose.
So I taped that up and went on a quick drive, no luck there either.
Speaking of the headlight air channels, I noticed that the driver's side has this rubber block in place and it's not there on the passenger side. Can anybody else please look to see if these exist in their cars?
I'm not sure what to test next, maybe tape cardboard over the intakes for both of the sets of slats in the front to at least try to localize the problem. The slats themselves kind of rattle around in their connection points on the sides and I could see the slats themselves potentially vibrate with the air pressure. I can feel the vibration in the accelerator when it's happening, especially at higher speeds so it would need to be something somewhat big.