You can't hose down mulch or ground cover plants so I would set the pavers or flagstone in river rock... 2-3" size...so you don't kick up too much gravel either.

Are your dogs currently comfortable doing their business on pavers? Or are they earth users? Many of the dogs I have known of prefer to use earth over hard surface. So if your dogs aren't currently used to using hard surface, try introducing them to it to see how they react.

If your dogs won't or don't use hard surface, you might want to consider getting a pet pad. It's like astroturf, so their pets can have "earth" to go on. You need to pick up solid waste, but liquid stuff can go through.

If your dogs will use hard surface. Don't use a wood mulch or live plants in there. Wood mulch will retain the stuff, and plants will die. Use a stone mulch, as already suggested. Be diligent in solid waste removal, and hose down the area often.

If you have cleanup issues, check into enzyme kennel cleaners.

Thanks for the comments. We will retain 2/3 of the backyard as lawn (natural turf); this is where I am hoping they’ll preferentially do their biz since it’s further from the covered patio, but want to be prepared for it if they go on the new permeable patio. Could I plant woolly thyme or some other ground cover in the river rock, so you think? Also, does anyone know if there are specific guidelines on how widely I would space the flagstone or pavers in order to meet permeability requirements? Again appreciate the thoughtful comments.

Is the permeable patio also going to be covered? If so, it might be too shady for thyme. A moss might work better. If it's elsewhere and gets some sun... here's a handy website for you. It's about the different ground cover thymes and how to best utilize them: https://mountainvalleygrowers.com/blog/groundcover-thymes/

It can depend a bit on what your soil is make of- clay vs loam vs other... how permeable it is to begin with. But in general, gaps between the stones of .5-3" is considered permeable.