There's not much out there that's tougher than St. Augustine. Lack of sun may prevent any turf from growing thick enough to stand up to wear.

Think about replacing lawn with another play surface in sections where grass is not holding up.

I’ve heard varying stories on how wear resistant St.A is.

What is the dark green looking grass? Did you overseed with rye or is that just something that came up? Bermuda would never survive in the shade of the wall and wall of plantings hiding the wall. No grass will survive continual beatings by pets or active children. However, a sod forming turf like St Augustine, bermuda, bentgrass, Kentucky bluegrass and sometimes zoysia can recover by itself unlike a fescue or rye. Fescue and rye are bunch type grasses which must be reseeded to recover from wear damage. So with shade your only two choices are fescue (no wear resistance) and St Augustine...or some alternate ground cover which likely would not tolerate children at all. Or you could go with mulch, which is kid friendly, low maintenance, and has to be replaced periodically depending on how deep you go with it. The bark mulch at my daughter's playground was 15 inches deep and very spongy.

That’s very helpful, thanks @dchall_san_antonio. The dark grass is some rye I put down a few months ago. I’m thinking at this point to encourage the St.A and just live with it. Our DeAnza Zoysia out front does decent in moderate shade which is surprising. Supposedly the Bandera Bermuda has moderate shade tolerance.

Any tips on getting the St.A to fill in?

Yes, see the previous post.

We had a similar situation when my kids were swing-set age and we had St. A grass in their play yard. The lawn, between kids and a little shade, looked awful for years. Our front lawn, St. Augustine as well, gets full sun and is tough as nails. Of course St. A is dormant now and will perk up in the summer. So maybe see if you are happier this summer.