Go with the shade mix, but don't expect too much in either case. Grasses...just don't do that well as light levels approach 3 to 4 hours a day and do tend to get patchy, so plan on re-seeding yearly or every other year.

Red fescue (it's green, not red) will tend to do best in lower levels of sunlight, with 3 the absolute floor.

"...has a large oak"

Danger, Will Robinson. This is a second challenge in that the oak is going to suck up the water and resources from the lawn. Areas near the oak are going to be difficult at best, and even further away, the oak has roots that are competing with the lawn. Very large, mature trees will often disallow lawns beneath them.

Allelopathy (the tendency for plants to poison things that grow beneath them) can come into play with some oaks and can poison grasses. No matter what you do, it just won't grow...because the tree is killing it to keep the space for itself. Without knowing exactly where you are and the type of oak, we can't say how strong this effect would be.

Also, depending on where you are within the grass range listed, you may be running out of time. Don't dawdle, would be the takeaway lesson here if you plan on seeding this year. Optimal time for most northern grasses to be planted was September first. It's...far past that point. Roots just don't have that much time remaining to establish before the colder parts of fall really get going.

Further south in the transitional range, you should still be OK, but also might want to consider a southern grass type anyway, which would be planted in spring. Since you're looking at northern grass types, you're probably on the more northern side of the range and definitely pushing it.

I’m located just north of Atlanta, it could be problematic with the trees, but I don’t know if a southern grass type would be a great solution. Most others in my neighborhood have fescue that is formulated for hotter temperatures and it does alright. There is some brown out, but it is already greening back up. I was planning to seed this week as temperatures are starting to cool.

Even though it’s 3-4 hours of intense afternoon you would suggest the shade mix?

With the Earth warming, you are just barely in the transition zone. I'd be looking at warm season grasses like St Augustine, hybrid bermuda, or possibly a zoysia (can't believe I said that). St Augustine will do the best in the shade. There are some zoysia varieties that do okay in the shade.

Agreed. Georgia really isn't transition zone any longer. It's southern grass. This year, my summer was barely Temperate/Transition Zone and I'm in friggin' Pennsylvania! :-)

3-4 hours will be the same whether it's around noon, afternoon, or morning, really. The intensity would be around the same except for fairly early in the day or close to sunset as far as the lawn is concerned, so there's no need to account for it.

Funny, my son was just saying today that he loves his zoysia lawn as we watched his two-year-old daughter running around barefoot. I didn't know that is what he had, but it really is a lovely grass.