This is extremely weird. I've searched Euripides (Middle Liddell's ref.) for every inflectional form of σκυθρωπ-, and can find no connection to puppies, even in the fragments.
The word σκυλάκαινα, in any inflectional form, occurs only once in all of Greek literature, in the 3rd-cent. BCE poetess Nossis (AP 9.604.3), who predicts a "watch-puppy's" (οἰκοφύλαξ σκυλάκαινα) amazement at the painted likeness of her mistress.
Why would such an obscure (at best) reference appear only in Middle Liddell, and not in the big LSJ? I must be missing something.