re: Lawn question - part shade and part full sun

St. Aug is your best bet in the shade.

A few things about shade,
Soil could be highly compacted like concrete because the shade maker is constantly sucking it dry. You could consider amending the soil.

The grass under the shade maker could suffer because of lack of water, because the tree will take it's share. Consider watering this area more than others.

You could consider thinning the tree to help with lack of sun.

Do weeds grow in that area? If weeds won't grow there, neither will lawn grass.