Palmetto is a type of St. Augustine that does well in the shade. The most common varieties of St. Augustine like Floratam will not grow in the shade. Palmetto will do fine in the sun too and it's said to be bit more drought tolerant. It will blend in visually with your other (Probably Floratam) St. Augustine grass.
john

There really is no grass ( turf) that does well under oaks.
Oak trees don't much like grass under them either.

From the link St Augustine For Florida Lawns
"'Palmetto'
'Palmetto' was a selection found by a Florida sod grower in 1988 and was released in the mid 1990s by Sod Solutions. It is of intermediate growth, with shorter leaf blades and internodes than many other cultivars, but is slightly larger than the dwarf St. Augustinegrass cultivars such as 'Seville' and 'Delmar'. It has a nice growth habit and does well in full sun or partial shade, but not in dense shade. It is sometimes referred to as drought-tolerant, but research has not shown that it has any greater degree of drought tolerance than other St. Augustinegrasses. It often has problems with disease, particularly in Florida's humid environment. It tends to have a lighter green color than many other cultivars."

The link below has good info too.

Here is a link that might be useful: Growing Turfgrass in the Shade

We had Palmetto installed under a huge Chinese elm years ago because it was supposed to do better in the shade. It lasted for about a year and fizzled. It didn't die out completely but it never looked good. The best thing for plantings under shade trees are shade-tolerant plants and ground-covers and heavily mulched beds. It's best not to fight nature, it always wins. I found that out after moving onto wooded acreage. We don't fight the trees anymore, I let nature plant under them. We had the trees and underbrush cleared out to 60 feet all around our house and planted lawn, but when it hits the woods it ends there, we let it take care of itself except for the twice-yearly bush-hogging.

Forget the grass. Bromeliads are good for covering the dirt where the grass won't grow.

That's just what I was about to say, forget the grass and make it a planting bed with bromeliads. :o) That's a great picture.