I have not used this, but have heard good things about these grass blends. Have the soil tested, and adding what they recommend (probably lots of lime)adding some compost (1/2-1") and having the area core aerated afterwards will help establish the grass. Fall is a great time to start a lawn in the south.

Dwarf mondo grass is a great idea, however, it will be expensive and it does not come in sod form.

Here is a link that might be useful: Shady grass blend

Mondo grass won't do what you seem to be seeking. I don't know of a "groundcover" other than a type of turf grass that meets the criteria as something for kids to play on. Now, you could consider whether you wanted to create a play area much like a playground, with bark mulch or some other product, in your yard, but there are maintenance, tracking mess, and other issues involved in those choices as well. Otherwise, I agree with Creatrix. If you can't do it yourself, you might see if there is a local lawn consultant that can analyze your situation--is the problem just shade, or tree root competition, compacted soil, pH, combination of all? These answers will influence what you can correct or not and what might grow there. Also, identify more specifically what you are trying to accomplish. How much room for kid's play do you need and what do they want to do that they can't do in the shady area? Playing football? Just trying to have some square footage that's not muddy? There may be different areas (sort of like micro-environments)--within the whole area you've got, so that maybe with planning you could have a small grassy area, a mulched area, a sitting area, etc.