Few, if any, turf grass varieties will tolerate such minimal sunlight. Even without the dog activity you would have a great deal of difficulty in establishing a decent lawn!!

While there are a bunch of groundcovers that are happy in shade or even prefer it, few are very tolerant of heavy foot traffic, especially if you (or the dogs) use routine pathways through the planting. Woodier types of groundcovers seem to hold up better than just the leafy, mat-like ones but these are usually slower to fill in and not as low to the ground. And you want to make sure whatever groundcover you choose is fully evergreen, otherwise the yard will become just a muddy mess in winter.

Since we have no idea where you are located, I have no idea what would be suitable for your area. In mine, I'd consider something like Cotoneaster dammeri (bearberry cotoneaster), Arctostaphylos uva-ursi (kinnikinnick), Rubus calycinoides (Taiwan creeper) or Gaultheria shallon (salal). All are quite shade tolerant and all are evergreen. And although I am reluctant to suggest such an aggressive and potentially invasive species (depending on location), English ivy, Hedera helix, would work very well also.

An alternative might be to define routinely used pathways/dog routes and surface them with wood chips. Then fill in the other areas with a groundcover of your choice (although I'd still lean to something evergreen).

Thank you. That is very helpful. Forgot to mention we are in zone 7, Arkansas. There is definitely one well defined path that the dog likes to take. I think I will set a path there, either with wood chips or maybe flagstone, and then fill in the rest of the lawn area with ground cover. I am definitely wanting something low growing that spreads quickly so we will still have the visual effect of a lawn instead of a bunch of clumped plants.

I'd certainly consider Liriope spicata, aka creeping lilyturf. This is a low growing, evergreen, grass-like plant that is very shade tolerant. And spreads pretty vigorously.