rAma did not slaughter or hunt animals and this was one of the main virtues of rAma highlighted by nArada when he was advising vAlmiki to compose story of rAma:
प्रजापतिसमः श्रीमान् धाता रिपुनिषूदनः |
रक्षिता जीवलोकस्य धर्मस्य परिरक्षिता|| १-१-१३
He equals Lord Vishnu, the Lord of all creatures, Sustainer of the Universe, together with Sri, Eliminator of enemies, Protector of the world of living creatures and complete Guardian of probity... [1-1-13]
vAlmiki also lists this down as one of quailities of rAma :
अप्रधृष्यश्च संग्रामे क्रुद्धैरपि सुरासुरैः |
अनसूयो जितक्रोधो न दृप्तो न च मत्सरी |
न चावमन्ता भूतानां न च कालवशानुगः || २-१-३०
Even enraged celestials and demons could not defeat Rama in battle. He had no jealousy. He conquered anger. He had no arrogance and envy. He had not even humiliated any living being. He had not surrendered to time.
वृद्धिकामो हि लोकस्य सर्वभूतानुकम्पनः || २-१-३७
मत्तः प्रियतरो लोके पर्जन्य इव वृष्टिमान् |
"Is not Rama, as a raining cloud to the earth, better liked by people than me, as he desires the development of the world and has compassion towards all living beings."
rAma was a follower of veda and the esoteric meaning of meat/ flesh as per vedas is either rice (aitreya brahmaNa - book 2-chapter 1) or rice+barley (shatapatha brahmaNa 22.214.171.124 - 126.96.36.199):
aitreya brahmaNa - book 2-chapter 1:
The chaff and straw of the rice of which it consists are the hairs of the animal, its husks the skin, its smallest particles the blood, all the fine particles to which the rice is ground represents the flesh and whatever other substantial part is in the rice are the bones
shatpath brahman 188.8.131.52 - 184.108.40.206:
1:2:3:7 It entered into this earth. They searched for it, by digging. They found it (in the shape of) those two (substances), the rice and barley: therefore even now they obtain those two by digging; and as much efficacy as all those sacrificed animal victims would have for him, so much efficacy has this oblation (of rice &c.) for him who knows this. And thus there is in this oblation also that completeness which they call 'the fivefold animal sacrifice.'
1:2:3:8 When it (the rice-cake) still consists of rice-meal, it is the hair. When he pours water on it, it becomes skin. When he mixes it, it becomes flesh: for then it becomes consistent; and consistent also is the flesh. When it is baked, it becomes bone: for then it becomes somewhat hard; and hard is the bone. And when he is about to take it off (the fire) and sprinkles it with butter, he changes it into marrow. This is the completeness which they call 'the fivefold animal sacrifice.'