Dogs are kind of individuals. Some love to dig, some rarely dig, some only dig if they are chasing
moles or gophers or voles. Some like to dig in grass, some prefer soft soil that you've already prepared for easier digging. So, sadly, it depends on the dog, and you can't really predict it in advance.
I've got six rescue dogs of mixed mutty breeds, and two of them dig like fiends. The rest aren't much interested, although they will help in a hot pursuit of a lively burrower. We have a lot of grass on the property in proportion to my plantings, so dog damage has been minimal. They did dig up and utterly destroy (by chewing) one nursery fig tree that I planted near our house. And one of them buried a bone in the metal half-drum where I store my potting soil, which was funny to find with my trowel.
But the thing to remember is, my dogs have acres and acres to roam if they want, most of which isn't planted in any way. So the
garden damage from their digging has been minimal. In your situation with a much higher proportion of garden to total area, it might be different.
However, any breed of dog will at least chase (and possibly catch) a wide variety of small herbivorous varmints. It's my opinion that your losses to dog activity will be more than balanced by losses you do not suffer because of the dog pressure on your varmints. My dogs eat several
rabbits a week, chase
deer when they notice them (not always at night if the dogs come inside), and also bring back (and sometimes eat) quite a few field rats and burrowers like gophers or voles. Just the other day my ninety-pound black behemoth dog (part Great Dane maybe?) was hanging around with me in the container garden when she suddenly jumped up and made a quick charge toward some of my pots. Then she came back very proud of herself and displayed a big fat very dead field
mouse. An unlikelier mouser you will never see!