To mitigate the spatiotemporal mismatch between renewable energy supply and building heating demand, this study proposes a novel non-pressurized shell-and-tube latent heat storage (NP-LHS) device coupled with an air-source heat pump (ASHP) system. To overcome the inherent low thermal conductivity of organic phase change materials (PCMs), the thermal performances of plain, corrugated, and finned tubes were systematically compared using both computational fluid dynamics (CFD) simulations and full-scale experiments. Numerical results indicate that the optimal tube spacing ratio ranges from 1.0 to 1.5. Among the evaluated geometries, the finned tube configuration exhibited superior comprehensive performance. It achieved an exceptionally high PCM volume fraction of 92.5% and dramatically reduced the complete melting time to 180 min—significantly faster than both corrugated (280 min) and bare tubes—while attaining a higher terminal temperature. Full-cycle dynamic experiments further demonstrated that integrating the finned tube NP-LHS into the ASHP system yielded a peak-shaving power reduction rate of 98.0%, effectively maintaining indoor thermal comfort. These findings conclude that expanding the conductive surface area via fins is practically more effective than inducing fluid turbulence for low-conductivity PCMs in non-pressurized storage applications.