I can't seem to stop shaking my head and smiling over this ... although I'm sure y'all are well beyond smiling over the mess! We went through this many, many years ago, with a long-haired cat. The horses pulled down the strip and the cat chased the pretty "ribbon" as it fell. It happened too quickly to stop, and poor Clancy was an instant mess! Once we caught him, it took hours to get the crud off him ... hours, a few judicious snips with scissors and a great deal of lard. Then came the baths to get the lard off. His one side looked like a greased pig for two weeks, poor baby. And holy cow, if you don't believe a cat can sulk, you missed a real show ... I wish I had pictures!
We solved the issue of disturbed fly strips by hanging them horizontally with a chicken-wire cover right beneath them. Worked like a charm!