Update:
Instead of fussing with a plane mesh, I tried using a full-screen dynamic texture as suggested by @RaananW here https://forum.babylonjs.com/t/how-to-make-a-gui-to-follow-the-camera-in-xr-vr/12259/5?u=owen
This works pretty well on desktop and I didn’t have to do anything to make the GUI stick to the camera.
However when I put on the headset, (Oculus Quest 1) the GUI is not usable… the left eye and right eye show the GUI elements overlay way too far apart for my brain to converge the overlay into a single image.