Asnes makes backcountry skis without metal edges especially for skiing with dogs. The website shows them in various widths and sidecuts. It looks like they are mainly used for hunting (with dogs).
Most of the time I'm fine on skate skis, even breaking moderate trails with the dogs pulling. When I go fast enough, even a skate ski floats! I've done fast BC single track and in very icy conditions with my two German Shorthaired Pointers pulling using my downhill setup, but I worry that the metal edge could cut the dog's leg or paw pretty badly. Although my control is excellent on this equipment and the dogs can really fly pulling me like a sled, poling and skating when necessary, accidents can happen.
Sometimes I take off the skis and boot up short steep inclines with the dogs pulling me in the harness. It's super easy even in Alpine boots. Sometimes the descents are too hazardous so I let the dogs run free and ski it solo.
It's a hoot to carve turns at speed with the dogs speeding so some sidecut is fun. At speed in variable or deep snow, stiff high boots are nice. So I think, for next year I'll expand my quiver and get dog-friendly Asnes edgeless skis with moderate width and sidecut. I'd choose 75mm cable bindings or perhaps minimal tech bindings (Dynafit Radical Speed Turn 2.0 Alpine Touring Ski Bindings) with heel lock. Boots will be T4's or very light AT. But my dogs are fast and I'm pretty light (100lbs of dog, 115lbs person) so that's the extreme control I need in some wildly fun conditions.
Just don't use metal edges with any frequency or something bad could happen. A helmet with goggles is good on these days.