Laser-controlled nuclear decay offers a new pathway for manipulating nuclear lifetimes. In this study, a novel scheme was proposed to indirectly manipulate α decay via electron screening in the scenario of laser-heated nanoclusters. We established a framework of the density-gradient-corrected screening model for the dynamically expanded cluster plasma, where the nonlocal effects on electron screening arising from gradient corrections to the spatial heterogeneity of electron density should be considered. These corrections eliminate the density divergence of the standard screening model near the nucleus. A quantum tunneling method on α decay was combined with Particle-in-Cell simulations of laser-cluster interactions to investigate the impact of extremely warm dense matter on α decay. The results indicate that the screening effects in warm dense cluster plasma can marginally alter the half-life by affecting the penetration probability. And this indirect laser-driven electron-nucleus modulation dominates over direct laser-nucleus coupling at low laser intensities. In addition, by varying laser and cluster parameters, we perform a comparative analysis of effects of these two mechanisms on α decay. It is shown that laser intensity is an important basis for distinguishing them. These findings offer crucial theoretical insights into the effects of electron screening on α decay in warm dense matter and serve as a valuable reference for future experiments aimed at laser-nucleus interactions.