I am a practising 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 practise 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
uma1966