Thanks for the kind feedback, markfug, very much appreciated.
Yes, the Background object (nor the Sky) has no idea about some of the channels in a material, nor that it would know what to do with lights.
I have attached a setup where the background object is replaced by a Plane. Since it moves with the camera and is excluded from all but the Ambient light source, it should work. If you chose any other field of view, the size needs to be adjusted. (With a 36mm lens and 36mm wide sensor the width equals the distance to the camera. Just to get an easy starting point.)
If you change now the color of the Ambient Light, it stays for both.
All the best