Composites of flexible and rigid polymers are ubiquitous in biology and industry alike, yet the physical principles determining their mechanical properties are far from understood. Here, we couple force spectroscopy with large-scale Brownian Dynamics simulations to elucidate the unique viscoelastic properties of custom-engineered blends of entangled flexible DNA molecules and semiflexible actin filaments. We show that composites exhibit enhanced stress-stiffening and prolonged mechano-memory compared to systems of actin or DNA alone, and that these nonlinear features display a surprising non-monotonic dependence on the fraction of actin in the composite. Simulations reveal that these counterintuitive results arise from synergistic microscale interactions between the two biopolymers. Namely, DNA entropically drives actin filaments to form bundles that stiffen the network but reduce the entanglement density, while a percolating actin network is required to reinforce the DNA network against yielding and flow. The competition between bundling and percolation triggers an unexpected stress response that leads equal mass actin-DNA composites to exhibit the most pronounced stress-stiffening and the most long-lived entanglements.