St Augustine will be the most shade tolerant of the two if it is really shady. Make sure you keep the tree trimmed. I have monster live oaks in front and back, and have a crew come out every 2 years. They do a good job of not making the trees look stupid, and it makes a big difference in my St Aug looking good

mondo or dwarf mondo. even the shade tolerant grass wont do well under a full tree.

Yeah, any sections that stay pretty much entirely shade all day even with trimming, it's time to consider a non-grass alternative. But with the right trimming effort along with keeping good care of the sod with proper watering and fertilizer, you should be able to really limit how much space that is

I will let you know at the end of summer. Two large live oaks on the north side of the house. Could never get any of the St. Augustine varieties to work because of the shallow oak roots. Went to Zorro Zoysia two years ago but it always struggled out of winter to repair itself from all activity from kids playing. Last fall I overseeded with Nightshift's Turf Type Tall Fescue and it has absolutely rocked all spring. Between the irrigation and no direct sunlight from 1-6, hoping this can hold on till the fall and become permanent.