Recent experiments have observed magnetic reconnection in high-energy-density, laser-produced plasma bubbles, with reconnection rates observed to be much higher than can be explained by classical theory. Based on fully kinetic particle simulations we find that fast reconnection in these strongly driven systems can be explained by magnetic flux pileup at the shoulder of the current sheet and subsequent fast reconnection via two-fluid, collisionless mechanisms. In the strong drive regime with two-fluid effects, we find that the ultimate reconnection time is insensitive to the nominal system Alfvén time.