If you don't need to walk on it, what about a groundcover like Geranium macrorrhizum, Big-Root Geranium?

If you're going with a groundcover, I'd choose something short that tolerates light foot traffic like violets or ajuga that you could occasionally mow over or skim with a weedwhacker.

There are grass cultivars for shade. I've found they don't really do all that well in shade. No grass does. They just do better than other cultivars.

Violets are probably native, but ajuga isn't. You could try Allegheny pachysandra or Canadian ginger. I have a three-toothed cinquefoil that colonizes my lawn. Green and Gold is recommended, too.

If you over fertilize grass growing in shade it will do much better. Your "forcing" the grass. I'd suggest if you want a lawn there to get a quality grass seed mix that's not from a big box store. You don't want to buy grass seed that will grow anywhere in the world. You want all the seed to be great in your area. Google Pennington seed and look for a dealer in your town.

Fescues are the grass that grow well in the shade. But no one ever points out that they also don't do well in drought conditions. So they won't do well if the shade is from a huge shade tree. You want a mix with perennial rye, a fine fescue and also Kentucky Blue Grass seed in the mix. If you have all fescues or Blue Grass or only one of them you'll have to dethatch your lawn which is a big job. Also those two varieties tend to get Red Thread and Red Dust when the humidity is high in mid summer. With the mix of all three you eliminate both problems.

I spread the seed by hand. Making sure I get an even spread. Take a big handful and throw it. You can only see the big rye grass seed so you'll have a lot more coverage than what you see. Be careful around garden beds to not get seed in them which you'll have to weed out. I'd suggest you cover the seed with an inch of "mushroom compost" which may be sold in bulk by the seed dealer. Some sellers will let you fill big contractor trash bags which you can put in your vehicle. Otherwise it's sold by the cubic yard. You could also cover the seed with peat but that tends to float away if there's any water flow thru the lawn.

Using the mushroom compost will improve your soil and give you a good start to over fertilizing you shady spot. I normally don't recommend fertilizing where you spread the mushroom compost the first season but you want to keep the shady spot well fertilized so you might apply a smaller amount than recommended in the fall. You can plant the seed late in winter and when the temps are favorable the seed will germinate.

The simple fact is that grass just does not grow well in shade. Even very shade tolerant varieties need at least 4 hours of uninterrupted sun. Anything less and germination will be low, the resulting grass will be sparse and weeds and moss - far more shade tolerant - will proliferate, choking out or overwhelming the struggling turf grass. There is also the issue of lack of soil moisture, root competition and soil compaction.

I agree that a low growing groundcover will be your best bet. There are varieties that will be very shade and drought tolerant and most are able to support light foot traffic at the very least. Evergreen is best :-) Unless you prefer, it does not have to be a native species. Just make sure it will tolerate your growing conditions and is not considered an invasive species in your area.