We study nearly unstable bivariate cumulative heavy-tailed INAR($\infty$) processes and show that, under a one-factor parameterization and a suitable scaling, they converge to the rough Heston model. This yields a discrete-time microstructural route to the joint price-variance dynamics and gives explicit formulas linking the INAR asymmetry parameters to the leverage correlation and diffusion scale of the limiting volatility process. On the pricing side, we derive the exact finite-$τ$ transform recursion and reduce it, in the diffusive scaling regime, to a quadratic discrete Volterra equation. We then compare this discrete equation with the continuous fractional Riccati equation from the rough Heston model. Under an admissible-strip assumption and local-in-frequency bounds, we obtain weak-error estimates for the truncated Carr--Madan pricing functional on bounded frequency windows of the form $C_1τ^{-α}+C_2(α)τ^{-(1-α)}$, where the second branch comes from the discrete-to-continuous Volterra comparison. The coefficient $C_2(α)$ collects the vanishing contributions arising from both the weakly singular baseline quadrature and the discrete-to-continuous resolvent comparison, and satisfies $C_2(α)\to0$ as $α\uparrow1^-$. We also develop an FFT-accelerated CDQ simulator with $\mathcal O(τ\log^2τ)$ complexity per path and use it to price European and path-dependent options, examine the classical limit $α=1$, and illustrate implied-volatility diagnostics.