Yes you are asking too much. I'd elaborate but I'm feeling very sleepy. Back tomorrow.

The best you can do in a shade area is to have a thin lawn. The only dense grasses in full shade are zoysia varieties. Those are very slow to recover from wear and tear from people walking on them. Dogs will rip them up and never let them return.

Having said that, there are some ground covers that look like grass but are not true grasses. Monkey grass is one of those. There is a dwarf variety that looks much like grass. Most of the ground covers I am aware of are not fast to recover from damage, though.

One alternative you might not care for is to cover your yard with several inches of shredded bark mulch. The dogs will love it and it is not going to be muddy when it rains. Even with two dogs it should never smell like a dog yard because of all the organic critters living in it. Two inches would be a minimum with 4-6 or more preferred.

I agree, you are asking for the impossible. The toughest grass that stands up to traffic and repairs itself extremely quick from damage is Bermuda. But it is full sun only and will not tolerate much shade as it needs 8 hours minimum of direct sun light.

Zoysia is tough and can take some light shade, but is extremely slow to recover from damage, like dirt slow.

Saint Augustine can tolerate a lot of shade, but cannot stand up to traffic and slow to recover...