Driveway conversion to dog play area
Have you considered artificial turf?  They have some now that looks like real grass, and is relatively affordable.

Mulch is bad because bugs can live there, and the dust will make it somewhat "muddy" when wet.  Well, maybe not muddy, but enough for the dogs to leave footprints in the house.  And there's no good way to clean pee out of the mulch, it will move or wash away with rain, etc.

The quality artificial turf will last for years and can be cleaned with a water hose.  You'd have to pick up poop, but I'm assuming you would do that with mulch anyway because... yuck.