As above metal oofing will condensate tar paper is fine. Our house is a similar size very different climate no snow maybe a couple of frosts a bit higher off the ground is nice when you want to go underneath to move or add a power point.
We clad ours in sugi weatherboard burnt and oiled with tung oil have a few practices some I didn't burn enough and the wasps love to eat it. Any uv or water will destroy tar paper in quick order.
What rafter spacing are you planing 1 in battens won't span very far