rhs=With[{},FullSimplify[
RHS - (1/2) b ϵ (ν^2) (2 (δ^2)+4 α β δ ν+(α^2) (β^2) (ν^2))RHS+2 ϵ ν(δ+α β ν)U[c]**RHS-β (1+2 b δ) ϵ (ν^3) (2 δ+α β ν)U[u]**RHS+α ϵ (ν^3) (2 δ+α β ν)RHS**U[w]+2 β δ (ν^2) ϵ U[c,u]**RHS+2 α δ (ν^2) ϵ U[c]**RHS**U[w]-(1/2) (β^2) δ (2+3 b δ) ϵ (ν^4)U[u,u]**RHS+(1/2) (α^2) δ (2+b δ) ϵ (ν^4)RHS**U[w,w]-2 b (δ^2) ϵ (ν^3) (δ+α β ν)U[u]**RHS**U[w]+2 (δ^2) (ν^2) ϵ U[c,u]**RHS**U[w]-β (δ^2) (1+2 b δ) ϵ (ν^4)U[u,u]**RHS**U[w]+α (δ^2) (ν^4) ϵU[u]**RHS**U[w,w]-(1/2) b (δ^4)(ν^4) ϵ U[u,u]**RHS**U[w,w] /. {NonCommutativeMultiply|U->Times, RHS->1}
]]

Collect[rhs, {c,u,w}, C[Simplify[#]]&]//Expand

Simplify[rhs/.ϵ->0]

Λ=(Collect[Simplify[Coefficient[(rhs/ν),ϵ]], {c,u,w}, C[Simplify[#]]&]//Expand)/.C[x_]:>x

Λ // TeXForm