Networked low Earth orbit (LEO) satellite constellations enabled by inter-satellite links offer a promising path toward ubiquitous broadband non-terrestrial services. However, fast orbital motion induces frequent scheduling updates and handovers, while stringent on-board constraints (e.g., limited radio-frequency chains) tightly couple user scheduling with cooperative beamforming. This paper investigates handover-aware power-efficient downlink transmission in networked LEO systems under statistical channel state information. We introduce a two-segment frame structure that separates handover-related operations from user-plane transmission, and propose a power consumption model that captures both the switching cost of newly established satellite-user links and the reduced effective transmission window during handover. Using a hardening-bound ergodic-rate metric, we formulate a per-frame network-wide power minimization problem with joint cooperative beamforming and implicit scheduling under segmented quality-of-service constraints, per-satellite power budgets, and serving-cardinality limits. To address scheduling-induced combinatorial sparsity and nonconvex fractional rate constraints, we develop an iterative algorithm that combines a reweighted $\ell_2$ surrogate with a penalty-based relaxation and a fractional-programming inner loop, yielding a sequence of convex second-order cone programs. Simulations based on time-varying orbital dynamics with frame-wise serving-set evolution and maritime user data quantify the power-handover tradeoff and demonstrate consistent power savings and improved feasibility over non-cooperative and pre-scheduled cooperative baselines.