I am a practicing Hindu and Brahmin with a reasonable knowledge of Ramayana and some of its different versions.
As far as I know meat eating is not restricted to any Hindu Caste, while in general South Indian Brahmins practice vegetarianism. Bengali Brahmins eat fish, Kashmiri Brahmins eat meat.
Further Rama is a Kshatriya(Warrior Class) son of King Dasaratha. Hence I do not see any reason why he should be a vegetarian. Hence in my opinion, there is no issue whether he is vegetarian or not.