The problem with loose ground cover is the mud in wet weather and dust in very dry weather. And on a slope, erosion and paws might just tear up anything you plant. Instead of sowing grass seed, couldn't you lay turf squares on the slope? That way the grass comes ready established and you just have to keep it watered until the roots knit into the soil. This isn't cheap but if the area isn't too big it wouldn't be that bad.

You might have to keep the pets off for a while but you could maybe do in two stages, turfing most of the slope to begin with and then roping this off, leaving a thin strip at the side of the steps for the dog to climb up. Then do this strip afterwards when the rest of the turf has established.