I've attached an image that shows the overall idea. Since this is a great place for discussing projects, I'll ask for some thoughts on my ideas...
I'm trying to do this project fast and easy, so I wanted to use the tray that you see in the photo since it is already veneered all around. As you can tell, it'll be a tight fit in the opening even with the dustcover removed. The turntable has about 1/4" of clearance from above when the tray is sitting directly on the lower platform.
Most drawer slides are about 1/2" thick, so I figured I'd dado cut recesses into the tray to allow it sit lower on the slides (when mounted underneath).
Of course, I could attach sides to the tray and do a side-mount slide, or use a thinner tray, etc. Many of ways to skin a cat, but undermounting seemed to me like the easiest way to get the existing tray to mount really low.