People have been making vegetable derived dyes for wool for thousands of years, so why not for wood!
I think the reason that the lye color goes so deeply is because the lye is reacting with tannins in the wood, so I wouldn’t guess that it will necessarily carry the turmeric color with it, but I don’t know that it needs to as long as your mix goes deep enough to color the surface, you’re good.
Ordinarily, naturally derived yellow dyes aren’t very lightfast, but since it would seem that the turmeric is also being darkened by the lye, maybe it wouldn’t fade as badly.
The one thing I would consider is determining how to neutralize the turmeric/lye solution to make sure it doesn’t prevent good adhesion of your finish.
As mmwood said, there’s one way to find out. I’d love to hear about the results.