This work investigates a bifractional reaction–diffusion model incorporating generalized temporal memory, nonlocal spatial diffusion, and directional asymmetry. The governing equation is H D t α, β ; ψ ρ (x, t) = D x γ ,θ ρ (x, t) + f( ρ (x, t)), x ∈ R, t > 0, where H D t α, β ; ψ denotes the generalized Hilfer fractional derivative with respect to a kernel function ψ (t), and D x γ ,θ is the Riesz–Feller derivative of order γ ∈ (0, 2] and skewness | θ | ≤ min {γ , 2 − γ} . The temporal parameters satisfy 0 < α ≤ 1 and 0 ≤ β ≤ 1. We first analyze the linear reaction case f( ρ ) = − ρ , H D t α, β ; ψ ρ (x, t) = D x γ ,θ ρ (x, t) − ρ (x, t), subject to the fractional initial condition I 0 + (1− β )(1−α); ψ (x, 0) = ρ 0 (x). ( ρ ) Applying the Fourier transform yields H D t α, β ; ψ ρ ˆ(k, t) = −(Φ γ ,θ (k) + 1)ˆ ρ (k, t), Φ γ ,θ (k) = | k | γ e i sgn(k)θπ/2 , while the generalized Laplace transform L ψ gives s(1− β )(1−α) ˜ρψ (k, s) = (k). s α + Φ γ ,θ (k) + 1 ρ0 Inversion yields the explicit solution ρ ˆ(k, t) = ρ 0 ˆ (k)( ψ (t)) β (1−α) E α, β (1−α)+1 (−(Φ γ ,θ (k) + 1) ψ (t) α ) . For ρ 0 (x) = δ(x), the fundamental solution is obtained and expressed in terms of the Fox H-function. We then consider the nonlinear logistic reaction f( ρ ) = λ ρ − µρ 2 , H D t α, β ; ψ ρ = D x γ ,θ ρ + λ ρ − µρ 2 , for which exact solutions are generally unavailable due to spectral coupling. To address this, we apply the Variational Iteration Method, decomposing the equation into linear and nonlinear operators. The resulting iterative scheme preserves the generalized memory structure through the kernel ψ (t). Using the linear solution as the initial approximation, we obtain explicit analytical corrections that allow a systematic study of the combined effects of memory, nonlocal diffusion, asymmetry, and nonlinearity. Future work will address well-posedness, convergence analysis, and qualitative dynamics such as pattern formation and traveling fronts.