I find it very hard to beleive someone would quit eating meat before living in a forest. This seems somewhat illogical, especially for an expert archer.
I don't trust that if it says that, that's a ridiculous idea, that He would stop eating meat to go into the forest. Why would He do that?
Let's bear in mind there's a lot of different Ramayanas out there, so judging something like this cannot be done by quoting sastra.
It's logically plausible that Rama did not eat meat in the palace and also not in the forest, out of compassion. It's also possible that He hunted when He lived in the palace and continued when He lived in the forest, out of dharma, since He needed to be skilled at warfare. It's also possible that He didn't hunt when He was in the palace, and started hunting when He was in the forest.
To say that he was hunting in the palace and stopped when he lived in the forest defies logic and explanation.