How to pave a small area but get maximum drainage / permeability

TL;DR: How should one going about creating a hard, smooth surface that drains well, where slope can’t be used to achieve the drainage.

We have a small 1/10th acre city lot, and two large dogs. Their waste products had been wreaking havoc on our plants, so I constructed a small ‘dog toilet’. I dug a 4×6 foot box about 8 inches deep, filled it first with pea gravel, then a top layer of river rocks. I then trained the dogs to eliminate exclusively in there. While the plants and grass have seen an improvement, there are some problems with the area as it is now. First, the dogs clearly don’t like stepping on the river rocks, and secondly the rocks make picking up their droppings rather cumbersome.

I’d like to basically replace the gravel and river rocks with a sort of smooth paved surface. However, I want to maximize its drainage capabilities.
While a simple solution would be to just use entirely the pea gravel, the two beasts will definitely kick the gravel out all over the place while they’re running around in the yard chasing each other.

So the next option I was thinking was perhaps paving in a sort of cobblestone style surface. A local landscape company sells larger river rocks that each have a large flat side to them. I think using those with sand filling in the gaps would do a good job of creating the surface and providing good drainage. However, I’m unsure the sand will sufficiently hold the rocks together, both when the dogs are stepping / running across them, as well as handling cold Minnesota winter frost related issues. Is polymeric sand a good option, would that still permit a high rate of water drainage?