So from experience, I think that a run Friday morning will NOT take into account Sunday forecasted rain. It will only look at that day (Friday). So on Friday, the system will take Thursday moisture level, add predicted precipitation for Friday only, and if the Thursday level + Friday precip forecast keeps your above your bottom Allowed Depletion line, then the system will be pushed out to Saturday....and then it repeats.
So if I were you, and you don't mind letting your yard hang out for a couple days, I would place it into a rain delay for 2 -3 days to see how Sunday pans out.
FYI, if you go to your moisture graph, you can see how the system will predict out the next 2 weels. Just look up towards the top of the screen by the date and you can click the right arrow to go to next 7 days. I'd screenshot but I can't at work.