According to Skanda Purana, Nagara Khanda, Volume 17, Chapter 124, Sage Valmiki is Brahmin by birth too.
Formerly, there was a Brahmana named Lohajangha in Chamatkarpura. He hailed from the family of Mandavya.
Initially, he was a good person devoted to his family but once, there was no rain for a long time in Chamatkarpuar. So, Chamatkarpur was affected by famine and to fulfill his family requirements, he became thief/robber.
Then there is another excellent Tirtha named Mukhara where excellent Brahmana-sages had contact with a thief.
That thief attained Siddhi, thanks to the power. Later he composed the epic Ramayana and became well-known as Valmiki.