Dog eats dog sounds correct but it's not an adjective here. As soon as it becomes an adjective we remove s from eats, Why?
Adjectives in English don't change to agree with the noun like they do in some other languages.
So if a noun is used as an adjective, it will be the singular form.
I have three cars.
I have three car decals.
It's easy to get mixed up because possessive nouns can be used attributively as adjectives and will keep the
's. So this sounds like an exception in speech but isn't.
I have the three cars' decals = I have the decals that belong to the three cars.