Entanglement-based quantum key distribution (QKD) protocols, such as E91 and BBM92, offer strong information-theoretic security and are naturally suited for satellite-to-satellite QKD (SatQKD) links. However, implementing these protocols over long-distance inter-satellite free-space optical (FSO) channels poses critical physical-layer challenges that are not addressed in the existing literature. In particular, photon losses due to beam divergence, pointing errors, and background noise can severely degrade the key generation rate and quantum bit error rate (QBER), especially under narrow receiver field-of-view (FoV) constraints. This paper presents a comprehensive performance analysis of entanglement-based inter-satellite QKD, focusing on photon-level modeling and the impact of practical impairments. We develop analytical expressions for signal detection probabilities, background photon influence, multi-pair emissions, and QBER, incorporating key parameters such as link distance, transmitter tracking jitter, receiver misalignment, and photon pair generation rate. Simulation results reveal the nonlinear sensitivity of system performance to tracking error and FoV limitations, and highlight optimal parameter regimes that jointly maximize secret key rate while maintaining QBER below acceptable thresholds. The proposed model provides actionable design insights for reliable and efficient deployment of entanglement-based SatQKD systems.