Ok, just for fun, I was looking at this more today. I noticed the walk up list was available by 6am today. I kept checking every 20-30 minutes and saw it was full at 11:20.
I’m not close enough to check in, so not sure if it shows approximate time you’d be called, but if you’re wanting lunch, check in as soon as you wake up. If you’d prefer dinner, wait until after the park opens.