Eating non-veg is NOT prohibited in Hinduism. Actually, the concept of vegetarianism as an extension of non-violence (in the matter of eating) came to Hinduism from Jainism/ Buddhism.
In any case, non-violence is part of ‘yama’ (right practices) for a Hindu spiritual aspirant and consequently, vegetarianism is part and parcel of it. Most Hindu spiritual masters do encourage vegetarianism only.
Ramakrishna Paramahamsa, being a Bengali Brahmin was eating fish (since fish is not strictly treated as non-veg by Bengalis). During his early days in Dakshineswar temple, he was reportedly eating goat’s meat too coming from Temple kitchen as prasad (off the goats offered as sacrifice to Bhavatarnini Kali) but later years, he stopped eating it. However, he has said that sacrificing goat to Kali (and eating it as prasad) on specific occasions was permitted as per Tantra scriptures.
Since Brahmins by nature and nurture were more satvic, they easily adopted to vegetarianism, even though Brahmins too were non-vegetarians in the distant past, as we could see in Mahabharata. When Padadvas were in exile, living in the forests, several Brahmins were staying with the Pandavas and giving them company in the forest. Pandava’s priest Dhoumya wanted to give a grand feast to these Brahmins one day and he requested Pandavas to hunt lots of animals and bring them for cooking and serving to these brahmins. It was when Pandavas went for this hunting that the King Jayatratha tried to misbehave with Draupati who was alone.
Interestingly, at a later time, there is a conversation that Yudhishtra has with his brothers. He says that he had a dream in which the animals living in that specific segment of forest came and pleaded to him that their population was fast dwindling and prayed to him to save their kith and kin! Yudhishtra, then decides to move to another forest from there. So much of eating of meat by Yudhishtra and bros along with their team of servants and the brahmins giving them company!