Click here this thread should answer the question.

do understand.. nothing will be drought resistant for the first few years.. depending on size planted .. also.. keep in mind.. something that creates GOOD SHADE ... might EVENTUALLY limit what you can grow under it .... but that would probably be many years down the road ... a pic of the area.. might get some better ideas .... ken https://sites.google.com/site/tnarboretum/Home/planting-a-tree-or-shrub

Mimosa wilt has been reported in California, notably southern CA, though the lack of characteristic symptoms noted in the OP makes this less likely. I'd avoid any cultural changes that might accelerate the tree's decline. As for planting A. julibrissin anywhere where significant disease problems have been reported, it would be best not to give the tree a prominent location, and to accept that it might have a limited life span (someone living about a half mile away from us had a huge old tree in their front yard that went from magnificent to dead over the span of three years). I have two specimens of "Ernest Wilson" grown from seed started three years ago, planted in two out-of-the-way locations. Both survived -9F this past winter with virtually no damage. There are cultivars i.e. "Flame" and "Union: which are said to be wilt-resistant. As for invasiveness, in my part of Kentucky I don't see self-seeded trees growing along roadsides and in "waste places" as described further south.

What do you consider 'small'? Your list of choices may stay somewhat narrow but not at all small in stature. And narrow doesn't provide a high degree of shade relief, regardless of height. Planting in multiples is also likely to overwhelm what is essentially a tiny plantable backyard. Small trees for small spaces generally are best kept to under 20-25 feet - crabapples, redbuds, most dogwoods, Asian maples, hawthorn, some magnolias. The question is will these provide sufficient shade to be of value and will they spread too wide to fit in that narrow planting area? I wouldn't be overly concerned about the possibility of Verticillium wilt being present in the soil. It is in most soils but stays dormant until prompted into activity by a stressed or damaged susceptible woody plant. I've experienced VW where it has killed a susceptible species but completely ignored several others very closely adjacent. Just don't plant a susceptible species in the same location as one that was infected and died and you should be fine. tbh, the suggestion of a shade sail or awning or even a pergola if the budget allows makes the most practical sense.