Dogs sleep pretty soundly, but this is definitely a malaphor. It is a mash up of “slept like a log (or baby)” (restful sleep) and “let sleeping dogs lie” (do not instigate trouble). Nice mix up as it involves assonance (log, dog, and lie, like) and similar words in the phrases (sleep, dog). This beauty was uttered by John Costello, one of my roving malaphor reporters!
This is a conflation of “let sleeping dogs lie” (leave something alone that might cause trouble) and I think “don’t beat a dead horse” (don’t waste time doing something that has already been attempted). When you mix dogs with horses, and sleeping with lying and dying, you get this malaphor. This one comes from Richard Lederer‘s Anguished English: An Anthology of Accidental Assaults Upon the English Language, rev. ed. Wyrick, 2006.