If you can successfully grow Morus nigra in Southern Louisiana, that species of mulberry would fit your requirements well. It is slow-growing and even if let to grow unrestricted, will not approach the size of a the other major species (white, red, or Pakistan mulberry) save for dwarf cultivars like Gerardi. Also, the fruit of Morus nigra is not only generally considered to be easily the best of all mulberries, but is often cited as one of the very best of fruits generally. There does not appear to be much difference, if any, between the fruit of the various nigra cultivars.
However, Morus nigra is relatively intolerant of humid conditions and, when young, is very susceptible to fungal infection. There are scattered reports of people growing them in the Deep South, but the majority decline and die. To attempt one in Southern Louisiana would be a risky proposition.
This year, in North Georgia, I planted Noir of Spain and a couple King James (all Morus nigra cultivars). The Noir of Spain promptly declined and died. However, the two King James are so far doing well, and are continuing to put on new growth into the summer. However, I will be surprised if I am able to keep them alive indefinitely (we shall see).
Pakistan mulberries have a unique, large fruit that is reported to be very sweet and less tart. The trees are very vigorous and will become quite large fast. They are very susceptible to late freezes. I planted several in North Georgia, and they were all taken out by a relatively mild late freeze after their second leaf.
Silk Hope is what is doing best for me in North Georgia. It is believed to be a red x white (Morus rubra x alba) hybrid, and has the more flavorful fruit of the red mulberry together with the productivity and vigor of the white mulberry. Like Pakistan, it is very vigorous and you will have your work cut out for you if you want to keep it small. The fruit is considered to be very high quality for a non-nigra mulberry, similar to Illinois Everbearing, a cultivar that should be avoided in your area because of its susceptibility to popcorn disease in hot, humid climates.
As for dwarf non-nigra mulberries, the first variety to investigate is Gerardi. It is a true dwarf cultivar and can be easily kept as a small tree or bush. If planted in a sunny spot, it is incredibly productive and will likely produce a greater weight of fruit per unit volume of tree than 95% of other mulberries. It grows well throughout the South and should probably do well in Louisiana. Reports on fruit quality are more variable, with some saying that it is as good as the best non-nigra mulberries, and others saying that the fruit is inferior to cultivars such as Silk Hope, Illinois Everbearing, and Oscar. You can find many detailed reports here on this forum from more experienced growers. (I have two of them but the birds swiftly got all the fruit of my tiny trees this year.)
Currently, all non-astringent persimmons are Asian (kaki) persimmons, so Americans and American hybrids would be ruled out by your conditions.
If non-astringent is your goal, you have two major categories of kaki: PCNA (pollination-constant non-astringent) and PVNA (pollination-variant non-astringent). The first category is always non-astringent, and the second category is non-astringent if it is pollinated. The first category (PCNA) includes Jiro (commonly sold as “Fuyu”), Fuyu, Izu, and several others. Many report that the fruit quality is pretty much the same between all PCNA cultivars, and so people make their selections based on when they want the fruit to ripen. One of the newest cultivars is Suruga, which some have reported to be of better quality (although perhaps more susceptible to cracking), but it is one of the latest-ripening cultivars and needs a long season (which you should have in Southern LA).
PVNA varieties have the disadvantage that they require two trees (for pollination) and that your fruit will necessarily be seeded, but many advocates claim that the fruit quality is better than PCNA. Notable varieties include Coffee Cake, Giboshi, and Chocolate, the latter of which makes many male flowers and works as a pollinator). I’m not sure if they are worth the additional hassle, but my PVNA’s are just coming into production, so I’ll be able to make my personal decision on the issue soon.
All kaki are smaller trees and should be relatively easy to adapt to your size requirements with moderate pruning. Kaki on D. lotus rootstock will likely be smaller than those on native (D. virginiana) rootstock, but the latter is considered to be the best for health and vigor, especially outside of the West Coast.
If you have not tried some of the astringent persimmons, you might consider trying some of the fruit. Like you, I figured that I would only want the easier PCNA varieties and planted many, but then I later discovered that I liked the fruit of the astringent varieties better. My favorites of the astringents are Saijo and Giombo. Also, American and American-hybrid persimmons are worth considering – the flavor profile is quite different than Asians (richer and more complex) and many greatly prefer them. Most Americans grow quite large, but many of the hybrids are smaller and would be worth consideration.