You can make lawns with the same kinds of sedums that are used for green roofs although you’d need to research that as theres all sorts of sedums and some would work better than others for the ammount of light ands wear they’ll get
You can make lawns out of clover (and bees will love you)
I’ve camped in france on a lawn of mint – totes amazeballs
you can also make lawns with camomile or thyme.
As much as its seen as a scourge of the lawn – moss to my mind is far nicer than grass.
It may also be you’ve just got the wrong grass, so maybe you need to lay or seed grass thats better suited to the shady conditions. Or with it being a largish area – unless you play bowls or tennis in the garden maybe turn it into meadow with paths amongst it rather than keeping it all mown and then having it muddy up