The only truly "dog-friendly" solution is to hand pull (and mow, repeatedly, perhaps with a weed-whip).  Glyphosate, the ingredient in Roundup, would, of course, be toxic were
a dog were to ingest it.  However, it does not stay around for long and you'd apply it directly onto the plant, so unless your dog is into eating those ash trees, it should be safe.  Remember that Glyphosate will kill anything green and growing that it touches.  Read here:
https://extension.umn.edu/planting-and-growing-guides/woody-vegetation-control