I had a challenging room though not as bad as you describe yours. I ended up using ARC room correction. I used Dirac as well but I liked Anthems line of products better. Either one should help with reflections and bass integration, I found the latter more important to me as I am not a fan of boomy bass that vibrates your insides. Another thing that would help is rugs on the floor and upholstered furniture.
Edit: I didn't notice you mentioned spinning vinyl. ARC or Dirac used would be doing analog to digital conversions which is something you might not want.
Edit: I didn't notice you mentioned spinning vinyl. ARC or Dirac used would be doing analog to digital conversions which is something you might not want.