I'd use mulch. It's softer if she falls, it's easier to remove when she outgrows the swing set. Kids will usually try going down a slide head first or jumping off a swing. You don't want either activity in the vicinity of a stone paver. If your kids don't do that now, they might before they outgrow the playset.

Specifically look for something called 'playground chips' - wood chips that have been tumbled to remove splinters and sharp edges.

Cedar bark is soft and makes good soft mulch. Has to be the bark. Stays in place nicely. Grass will eventually grow though it so does not kill the roots which is handy when the path is no longer used as much.

I think the path is adorable, and all too soon tiny Crocs will no longer wear down the path. Wait until that time comes, and plant grass there.