Stay in Shinjuku or Akasaka?

My boyfriend (25) and I (26) and travelling to Japan September 27-October 11. Our route is Tokyo, Kyoto, Miyajima, Hiroshima, Osaka, Tokyo. We’re staying in Shibuya on our second Tokyo leg but are stuck deciding if we should spend some extra money and stay in Shinjuku or stay in Akasaka for $100 a night instead. Any advice would be greatly appreciated! Thanks!

by Formal_Caregiver683