Low earth orbit (LEO) satellite constellations are becoming a cornerstone of next-generation satellite networks, enabling worldwide high-precision navigation and high-quality remote sensing. This paper proposes a novel dual-function LEO satellite constellation frame structure that effectively integrating navigation and remote sensing. Then, the Cramer-Rao bound (CRB)-based positioning, velocity measurement, and timing (PVT) error and the signal-to-ambiguity-interference-noise ratio (SAINR) are derived as performance metrics for navigation and remote sensing, respectively. Based on it, a joint beamforming design is proposed by minimizing the average weighted PVT error for navigation user equipments (UEs) while ensuring SAINR requirement for remote sensing. Simulation results validate the proposed multi-satellite cooperative beamforming design, demonstrating its effectiveness as an integrated solution for next-generation multi-function LEO satellite constellations.