Assuming you are going the route of using a still image for the background: Don’t go lower that the target resolution.
Which means you would need to know what the framing will be and then ensure that it is always 4K for that frame area. So if the frame area is 1/2 of the width of the full image, the image would have to be 8K, if the framing is 1/3 of the full image, the image would have to be 12K (width wise), etc.