At least in summer there's no evidence of 'no dog' signage. Many loops, with a wide variety of conditions. Co Road 644 off the Echo Trail 17 miles from Ely. Also, the Heartland Trail near Walker, Minnesota is popular. Another nice place I have skijored is near Miners Lake in Ely, Minnesota.
People there seem more receptive to dogs on the trail.
The Lodge does not attract as many skiers anymore. The Gateway trail near Stillwater is skiable. Dogs are less unwelcome on the horse trail. Take the trail north from Highway 36. You can also catch the trail at Highway 96.