Wireless energy harvesting (EH) and shortpacket communications (SPC) are key technologies for enabling ultra-reliable and low-latency Internet of Things (IoT) networks. However, the performance of rate-splitting multiple access (RSMA) under finite blocklength (FBL) and energy-limited conditions remains largely unexplored. This paper develops a comprehensive analytical framework for a power beacon (PB)-assisted RSMA downlink operating over double Nakagami-mfading channels. Closed-form approximations for the block error rate (BLER) of common and private streams are derived in the FBL regime, along with high-SNR asymptotic expressions and achievable goodput analysis. Monte Carlo simulations validate the theoretical and asymptotic results, showing close agreement under various fading and power conditions. The numerical results reveal important trade-offs among blocklength, time-splitting ratio, and RSMA power allocation, and confirm that careful joint parameter optimization significantly improves reliability and throughput. These findings provide useful guidelines for designing energy-efficient and low-latency RSMA-based IoT systems.