plant winter grass. We have two big dogs and had the same problem and with our decks and the winter grass it has really helped. We live about 300 miles direct south of you.
Winter grass must be laid down on or shortly after Oct 15 and it must be laid down thick. We have a very small front yard its about 30 X 20. We put down 150 lbs alone each year on the front.
Every seed is one blade of grass.
Build some decks, then install temp fencing for this kind of winters to keep the dogs off the grass if needed, but then they do need to use the bathroom.
I think that this year you are out of luck. But planning for next will save you.
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