You will probably never get grass to grow there. You have a trifecta of issues. I sympathize because my dogs have pretty much destroyed my yard too. I'm sure your puppy sprints out of the door and the grass will always get torn up right in front of the patio.
I have seen where people put a gravel area in the yard where they want the dog to go. They usually add something like this. https://www.drsfostersmith.com/product/prod_display.cfm?pcatid=11109

They hose it off regularly to keep it from getting smelly. It's easier to do this with a male dog.

Pet dog=ugly lawn. I overseed every year and a few months later, it is a mudpit again. It is the price you pay for never ending devotion and love :)

Overseed in the fall again and help the dog find a new spot. I had to go outside with him and guide him to where I wanted him to go. But now, he goes on his own. He now goes on all the English Ivy on my fence line that I hope will die. Of course, it is alive and growing even with urine all over it several times a day.

This may help.

Build a dog run and put him there to do his business when you are outside with him he can play in the yard and when you see him getting ready to do # or # pick him up put him in the run and after awhile he will get the message that is where he goes. Put pit run gravel in the run so it is easy to clean.Leave the run gate open so he knows it is his space.

my dog doesn't eat dog food, I cook everyday for her. Never have this problem with my grass. Maybe is what your dog eats?

I've seen where people have laid down green chicken wire secured with landscaping staples and then planted grass on top. This keeps the grass from getting torn up when the dog takes off running. I plan to try this in the spring.

I love my dogs, but I'm not cooking their meals everyday.

If you click the link above, you'll see it has to do with the way the protein in their diets turn into nitrogen. It's the same as putting too much fertilizer on your lawn. It also says that this won't be as much as a problem if you don't fertilize your lawn, that way you won't have an overload of nitrogen.

Higher quality dog foods have proteins that are broken down better and don't cause this problem as much.

This time of year it's likely the paws are doing the damage, not the pee. I have 6 dogs and never see pee spots until we get into at least the mid 80's for temps and even then it's not bad at all.

Over seeded my Central Texas Bermuda lawn with Perennial rye grass in
2017.

Overseeding with rye is a rookie mistake, but has nothing to do with your problem.

The ROI was lovely and was filling in nicely despite my particular
lawn specific issues of irregular shade,

Not sure what ROI means, but irregular shade is a problem for bermuda. It needs a good 6-8 hours of REGULAR direct sunlight in the summer in order for it to thrive.

Flash
forward to January 2018 when I brought my 8 week old Labrador retriever
home...

BINGO. Until the dog is 2 you can give up the idea of having a nice lawn right at his landing pad. Labs get fat and lazy later and then you can have a nice lawn. So far he has killed the rye (you will thank him for that) but he has also torn up the bermuda. That, combined with the shade, will seriously delay the return of the bermuda. As gardengal mentioned, the problem is excess urine that your soil is not prepared to decompose. This will sound crazy but try it...scatter a handful of table sugar onto one of his newest yellow spots. Water that down without washing it away and wait 3 weeks. The rye should return to deep green. What the sugar does is stimulate the soil biology to reproduce. Once you have a larger population of microbes in the soil, they can deal with the excess urine. You might be able to fix this by using organic fertilizer like alfalfa pellets (20 pounds per 1,000 square feet), at least once a year and spraying with molasses (3 ounces per 1,000 square feet), at least once a year. Then when you water go a full inch deep every time. This time of year you should be watering once a month, but that is fixin' to go to once every 3 weeks when the temps get into the 70s every day. You're going to get 3 days of heavy rain real soon (tomorrow). You won't need to water again until at least April.

Our dog parks in San Antonio use bark mulch as a surface. In some parks it is a foot deep. It is easy on the dogs' feet, doesn't track mud into the house, and allows the urine to go down deep where the microbes can take care of it. One of the best I know of is "Cedar" mulch out of the Hill Country. This is a great mulch anyway and gets rid of the cedar for the Hill Country ranchers. The plant is really a juniper, but anyway it makes great mulch.

Had a black lab that in the winter didn't like the cold and went just outside the back door in the lawn. No matter what I tried I could never get the grass to grow.

I can't see too much of your backyard. Is it that small? I think it's kind of a trade-off, the unconditional love and joy that a dog brings to a family (or single person). Or a low, somewhat successful ground-covering plant. Your photos show a lot of shade, and it may be tough finding the right type of sturdy grass for your area and growing conditions. And it just might not be worth the trouble. I personally wouldn't invest a ton of money in growing grass, if it doesn't seem like it will work. I definitely wouldn't put chicken wire with ground staples over grass. First of all that will rust. That is bound to result in damaged paws and vet bills for your dog, and/or for anyone who goes outside barefoot or in flexible-soled shoes. Those staples eventually come up. And how would you mow the lawn knowing the edges of chicken wire may come up into the mower? Yikes.

There are lots of alternatives to grass. Decomposted granite; other, sturdier types of ground covers (for your specific growing area); artificial turf; deck material, etc.

Finally, as someone pointed out, it is likely that the rambunctious phase will be over all too soon.

I would put the dog's well-being over the grass's well being.