I love my four cats, and of course they LOVE my quilts and feel it is the sworn duty to sleep on them to break them in right. I have several large white flat sheets that are used to cover quilts when I walk away from them. If it is in the machine, I would cover the entire machine, quilt, table, etc. The screen door to the sewing room sounds like a fabulous idea for you, but the cats are mine and would be terribly insulted. The batting can be salvaged with piecing. Claw rips are not so easily repaired.
God bless!