Reference existing WebGL depth buffer when rendering a new ThreeJS scene - three.js

Question

I have an existing WebGL canvas that is being rendered without using ThreeJS, and is for all intents and purposes a black box to me, apart from two facts: (1) I have access to the underlying webgl canvas DOM element and can position and resize it on the screen, and (2) I know the properties of the camera for the scene, and get updates on every render cycle for that camera.
The problem I need to solve can be simplified to the following: I need to have my own separate ThreeJS canvas that displays both the black box canvas data, and then elements that I draw, like a cube for a simple example. I can already easily overlay the two canvases, set the transparency on my canvas for everything but the cube, and then align the two with the camera events from the black box library. This works quite well.
The issue with this is that when I draw my objects, like a cube, they don't respect the depth buffer of the black box canvas. So I might have a cube that is properly aligned with the backing scene and movements of the scene, but then it isn't properly masked when something in the black box canvas is closer to the camera than the cube. My thought is that I need to solve this in one of two ways: (1) I can have my renderer write to the other canvas with autoClear = false and preserveDrawingBuffer = true, or (2) I can somehow copy the depth buffer from the black box canvas into my canvas, and then set up my renderer so that it respects the new depth buffer.
I haven't been successful with either approach yet, so I'm wondering if this is possible, and if so which of the above approaches, or what other approach, can solve this problem?
--Edit--
See https://jsfiddle.net/zdxyoajb/ for angular/typescript implementation of the above attempts. In the following animate loop, if I comment out the overlayRenderer lines, the below sphere will be red and offset from the center (as it should be), but if I don't comment the lines, I get the below image. I also get the following error:
WebGL: INVALID_OPERATION: uniformMatrix4fv: location is not from current program
animate() {
requestAnimationFrame(() => this.animate());
this.blackBoxCamera.copy(this.overlayCamera);
this.blackBoxRenderer.render(this.blackBoxScene, this.blackBoxCamera);
this.overlayRenderer.state.reset();
this.overlayRenderer.render(this.overlayScene, this.overlayCamera);
}