Ideally your ceiling would be very high for this setup to give space for the light to diffuse from hanging ceiling reflectors, which is unfortunately not the case for most homes. For realistic homes, there is a style of floor lamp termed 'torchiere' that may work, though for best results you're going to want to find one with a large, fully reflective shroud - lights this bright are pretty harsh if they're not diffused well.
Set up something like this on an automatic timer to simulate the sky, and I suspect a lot of our sleep issues would go away fairly quickly.
I had a similar hypothesis, so I built something like what you describe:
https://github.com/wpietri/sunrise
Currently I'm at only 4000 lumens or so in my main space, and I'd like it brighter. But my sleep is definitely better. I attribute it not so much to the daytime light, but to the way my apartment gradually gets dimmer and redder in the evenings.
I had set out to make something like a SAD lamp for my whole apartment. But as a side effect, I also basically made f.lux for my apartment, and I really like it. When evening comes, I generally have to fight to stay awake rather than fighting to go to sleep.