When you develop one, let me know. I will partner up with you by infusing cash and we will both be extremely rich and retired in Hawaii on our own private resorts. Might even just buy one of the Islands and make a new country. We can call it Best Weed Around.

Honestly of the warm season grasses SA is the most shade tolerant of all of them but does not stand up to traffic. SA variety does make a difference. Palmetto has the about the shade tolerance, and Floratam has about the least shade tolerance of SA grasses. If I had to guess the most popular varity in Houston is Florata. Palmetto is widely available in Houston if you would like to try, but sounds like a lost cause to me.

The most traffic tolerant is Bermuda followed by Zoysia. However neither tolerates shade worth a darn. You might try Celbration Bermuda as it is the most shade tolerant of the Bermudas followed by TifGrand. But again it sounds like you have deep shade which means no grass will grow there.

There is one shade tolerant Zoysia you can try to plug but it will cost you your first born called Shadow Turf. Last time I looked a flat of 72 2" x 2" plugs sells for $90. That is enough to cover 4 feet by 5 feet and will take 2 seasons to fill in if it lives. I have not heard much of anything good about Shadow Turf.

I have a small area of Shadow Turf. My biggest problem with it is the area is too small to get a mower into. Maintaining it as an ornamental area is more work than I'm willing to put in. I have seen it look incredible, but not with dogs. As TW said, dogs and grass to not work together. Add in shade and it gets worse very fast.

I usually suggest covering the yard in deep mulch for dogs. They did that at a local dog park, and it is very nice. Anything that starts to grow in the mulch is easy to pull out because the roots cannot get much of a grip.