well as it goes i have 4 dogs and a raised concrete block bed thats renderd and painted etc . it also has the benefit of being user friendly . you could use bricks or half blocks and do circles to.there reasnably easy . you just bang a piece of wood in the ground and tie a piece of nylon or strong string to it half the length of the width your after.then just lay the bricks,blocks or fence posts at the end. i wouldnt use wood as its a lot of work to rot out.you could keep it formal and have just concrete paths between your beds.you could have taller ones within shorter ones.the worlds your oyster realy.you wont notice the odd axcident with a few beds or have to cut the dogs nails on concrete and ofcourse it hoses down great.i have know grass thats almost bread into us to have a garden with grass.theres so many interesting things to plant than grass.
you can put trellis in it afterwoulds. i would just look at your garden in a different light perhaps. the other option is a dog run which possibly isnt practicle

theres quite a few blogs of my garden if not look under nosey potter on utube maybe .take care bye for now .