lets go with instead
Alpha (“the upper class”) Beta (“the second upper class”) Gamma (“the middle class”) Delta (“the second lower class”) Epsilon (“the lower class”) Double-Plus (“the superior subdivision among Alphas”) Plus (“the superior subdivision among Alphas, Betas, Gammas or Deltas, but inferior to Double-Plus”) Minus (“the inferior subdivision among Alphas, Betas, Gammas or Deltas”) savage (“a person outside the integrated portions of society, and therefore separate from all classes”)
Your point about a society capable of large scale gene editing would not need a working class is legit. it will be used to thin the populations out to 'manageable' numbers. Then class split based on parentage.
A perfect system and within 3-4 generations completely normal