How much access will you have to the eventual fruit in this situation? Fruit can get pretty messy when it hits the ground from a distance.

What about a sweet cherry tree? On Mazzard rootstock they can grow up to 35ft tall.

If you have the patience you could grow a sweet cherry own-root. This is virtually never done because it can be 9-12 years before you ever see any fruit, but the tree will be much bigger and stronger, and have a thicker trunk like a real tree. They really are a magnificent (and rare encountered) sight to behold.

Sweet cherries can grow to be one of the largest fruit trees.

I've heard sweet cherries suffer from disease and insect problems here in the muggy southeast, and I don't want to be battling those constantly.

Yes, I know that actually getting the fruit will be a problem. I figure my kids or I could work out some combination of ladders, tree-climbing, and those fruit-picking tools on long handles.

My other thought was a nut tree, but I've heard that it's not just black walnut, but other nuts as well, that hinder the growth of neighboring plants, and this spot is right near my vegetable garden. Correct me if I'm wrong about that.

It is true that cherries are generally more vulnerable to disease in warm muggy climates, but there are several varieties which are considered more disease resistant:
http://www.raintreenursery.com/fruit_trees/cherries/disease_resistant/

The loud,

I am in the east coast. If I were you, I would not try sweet cherry. Too many problems even those advertised as disease resistance. I do not know if sour cherry would do well down south.

if you want tall fruit trees, pears may be the best choice. You need to find southern pears that are fire blight resistance.

paw paw and persimmom are mostly problem- free trees but I do not know how tall they are.

Pears may not grow narrowly. Varieties such as Ayers, Orient and Moomglow are reputed to be disaese resistant. In a standard rootstock which you need to grow a very tall tree. it can take up to 10 years to fruit, though.

If A. Persimmon can grow tall, that would be a good choice. It is pretty much trouble free.

it is true, too, that if you cannot harvest fruit in time, dropped fruit can be quite messy.

Red mulberry