Just as Logos eternally applies to Christ/Son, so monogenes ('only begotten' weak KJV translation) is eternal with Him. Verses about being begotten (not monogenes) in other contexts relate to His resurrection (words have a semantical range of meaning, especially when they are different underlying Gk. words). The Johannine use of monogenes is different than the Pauline or Lukan use of gennao (and its roots).
You are guilty of an exegetical fallacy, assuming the same Gk. word always means the same thing or that the same English word always has the same underlying Greek word.