I can't think of a grass that will do what you're asking. You might consider letting the back yard go wild and see what does grow, if anything. If you did not have shade I would suggest a blend of Kentucky bluegrasses.

Dog urine and waste are not a problem in normal lawns. If you do a search here you will find some discussions, but, all things considered, not very many people have a problem with that.

Would you consider using a deep mulch instead of grass?

Tall fescue sounds like the answer. It's relavtively inexpensive, readily available for overseeding, something you will be doing once or twice a year with your dog and climate situation.
The grass you are thinking of that grows fast in blends is called perennial rye.
I would skip that and go with just tall fescue. Keep it very well mulched and watered religiously and it will germinate in week.

It's a pretty decent sized yard so it's not like they're concentrated in a 10'x10' section so it's not one particular are that they will be in, except for right off the patio where we let them out. Our back yard is approx 70'x40' I know it's not impossible to have a yard and grass, our previous house had a really nice yard with the dogs. The only spot that couldn't keep up was along one side where the border collie would pace back & forth searing the neighbors yard for squirrels, but I don't think any grass could take that... The one thing that yard had over this one was it was a little bit bigger and A LOT of sun. There weren't any trees back there so it was constant sunlight.

> "Dog urine and waste are not a problem in normal lawns"

With male dogs perhaps, because the male will mark many spots. A little here, a little there, a little everywhere!

However, a females urine is more concentrated because she tends to hold it longer and then only go once and in a single spot until her bladder is empty. This creates a strong ammonia or pure nitrogen, which then causes a burn spot. True dog urine spots will many times be brown in the middle and growing vigorously around the brown spot due to the nitrogen in the urine. It was just too strong in the middle.

Having said that, owning females is easier for the lawn lover. One can make a potty area. A small designated area, fenced off which is off to the side works well for females. They will adapt to this easy and will use this as their area. Pea gravel works very good for this. It prevents the area from becoming a mud pit and is easy on the pads of the dogs paws. Makes picking up of the stools easy too.

As for the romping around and seed selection, think big dogs playing professional sports. Baseball, football, soccer players are all hard on the their fields. Most fields are a mix of Kentucky bluegrass and rye. Rye will take a beating and abuse. An aggressive KBG cultivar such as Northstar, P-105 or Emblem, spreads and fills in any voids/damage created by activity but this is all dependent on proper irrigation and feeding. Failure to irrigate will only give you a straw field in summer, which the dogs will tear up in an instant. Then when it rains, they will be romping around in mud.

It can be done, even with a small yard - especially with a female.

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> "Dog urine and waste are not a problem in normal lawns"

With male dogs perhaps, because the male will mark many spots. A little here, a little there, a little everywhere!

However, a females urine is more concentrated because she tends to hold it longer and then only go once and in a single spot until her bladder is empty. This creates a strong ammonia or pure nitrogen, which then causes a burn spot.

I understand all that. I'm just saying that it doesn't seem to be much of a problem based on the relative infrequency of posts about it here. If your SOIL is pretty healthy, dog urine does not seem to be a problem. On another forum someone suggested scattering sugar on the entire lawn every year. I would agree with that.

It will be very difficullt to even get some semblance of an established lawn. Young grass seedlings is very delicate. Should not be walked on, played/romped on for a couple of months.
Mid june and temps in 80-90- forget about it. The shade will help for that but for germination and establishment it will be a pita. Recommend sod instead.

You might want to visit your local ag store to see if they have native grass seed. I found that works best. You may end up with clover and some weed seeds mixed in but by far is the best and will reseed itself. I too have three dogs, one is a female and except for occasional urine spots I found flushing those spots in hot weather really helps. There is only one area that I usually have to reseed and that's where all the dogs take off running after squirrels and that's right where the patio ends and the grass starts. We have about 1/3 acre of yard and to the rear is a wooded lot so we get both shade and hot afternoon sun in the summer. I do not mind having clover because it helps to keep the yard green and it's short, plus the clover flowers feed the honey bees in the spring. We live in PA so our seasons and temps are pretty much the same as yours. If you don't have an ag store near you, you might want to check online to find your native grass seed there. By far I found native grass seed to be more hardy than any other specialty seed, and believe me through the years I've tried them all and in all types of combinations. So if you don't mind some clover and a bit of weeds mixed in with your grass native is the way to go. But if you're looking for a lush weed free yard then good luck finding something you need to stand up to your specific needs with the dogs, and shade/sun and if by chance you find something please let me know, I'd be interested but only if I didn't have to keep reseeding often. I'm more interested in doing less work not more. Hope this helps.