Neutral atom arrays have seen exciting progress as a platform for quantum computation. However, as we move towards the regime of fault-tolerance, the large-scale impact of fundamental features in these systems is not well-studied. In this work we point out that the use of movement in neutral atom arrays may set an unavoidable constraint on the speed of computation, erasing potential quantum advantage. As one solution, we propose a movement-free QEC architecture based on groups of interleaved surface codes. Our architecture enables fast, high-fidelity transversal CNOTs on surface codes in the same group. We also introduce interleaved lattice surgery to create high-capacity routing channels between groups. We validate our architecture through detailed numerical simulations of the underlying circuits and we evaluate its scalability through compilation of key benchmark applications. In regimes of high parallelism, we find our architecture leads to a $\sim 3 \times$ reduction in compute time. Our architecture leverages experimentally demonstrated dualspecies atom arrays which exhibit asymmetric interaction strengths that scale with $1 / r^{3}$ for interspecies interactions and with $1 / r^{6}$ for standard, intraspecies interactions. We examine how such scalings enable interleaving with high fidelity and propose how error rates required for QEC could be achieved. We also evaluate the tolerance of our architecture to two-qubit gate fidelities. We find the advantage of interleaving admits sizable tolerances of $\sim 1 \times$ to $3 \times$ increase in error rates. We conclude the benefits of our proposed interleaved architecture grants strong motivation for future experimental efforts targeting longer range dual-species gates.