"Fuck the dog, for its part, goes back to at least 1935 with an earlier, more polite version, feed the dog, dating to the 1910s and meaning “loaf around.”

https://www.dictionary.com/e/slang/screw-the-pooch/#:~:text=Fuck%20the%20dog%2C%20for%20its,as%20World%20War%20II%20slang.