Aw, I am sorry. What a lovely idea that is. I live on the East Coast, so I can't help with any suggestions. Maybe check your state's with native plant society. Every state has one and they can give you good recommendations. I did see that dogwood trees will grow in eastern Texas, but they go get taller than 6 feet. Most trees do.

5-6 is not a tree, it’s a shrub. (Which is fine!). Things like pittosporum are often kept small and bushy but make great small trees with fragrant blooms in spring. You just have to limb them up as they grow.

Yes, look for a shrub that can be trained into a tree form. Hydrangea paniculata is an obvious choice but other shrubs are commonly trained this way also - dwarf crape myrtle, lilacs, ninebarks, etc. Or just go for a compact flowering evergreen shrub, like Texas sage, Leucophyllum frutescens

Vitex, chaste tree, is lovely, and I think Jilly has one in TX. might need to prune it eventually.

Sorry for your loss. I was going to say a Texas mountain laurel, but they get slightly taller than 5-6 ft. I had one I was pruning as a small tree when I lived in the Fort Worth area and it smelled wonderful and was very pretty.

Or how about a dwarf redbud tree. I’ve always loved them with their heart shaped leaves and gorgeous flowers.

Sorry about your pup.

For Houston a dwarf crape myrtle would be great. But you must likely would have to order one to get a true dwarf. In fact, same for most dwarf cultivars.

Or for a shrub, an azalea, if you can water. You have said sun, shade, & do you have a garden hose to reach the area.

Would you be browsing local nurseries or relying on big box? Most box stores have very limited inventory in fall. More independent nurseries will have a variety of shrubs though usually in 3gal+, so , not sure exactly how big a home you wish to dig; but you’ll be able to see shrubs of many types. but also they may not carry as many dwarf trees either. Whereas online you can find not only dwarf cultivars, but smaller ( younger) plants.