No to CLF! It is the number one item that lowers property values. It is industrial, not residential. The only residential use for that fencing is dog pens. Repair and paint the fence that is there. It should last until you and the neighbor can afford to go together and install a taller version of it. Invest in a Mantis tiller, or rent a tiller at the hardware store. Till up the backyard and scatter native wildflower seeds that thrive in Montana. That should hold you until you can make a complete plan and have the $$$ to do it right.

Agreed, you must reconsider the chain link. If you invest in a beautiful privacy fence, your budget for plantings can be dramatically reduced.

Once your fence is up I suggest planting columnar trees along the back fence. Unless you like looking at the red house. Trees are expensive. That may be all you can do this year.