We present a new electrostatic particle-in-cell (PIC) algorithm that remains accurate and stable for timesteps far larger than the particle gyro-period, in nonuniform magnetic fields [1,2]. The algorithm extends earlier electrostatic fully implicit PIC implementations [3] with a new particle pusher [4], accelerated by a fast Picard nonlinear solver [5] that enables the large-timestep regime without sacrificing the discrete invariants that underpin long-time fidelity. The key idea is a drift-consistent integrator: as the timestep becomes large relative to the gyro-period, the method preserves the correct averaged guiding-center drifts, while in the small-timestep limit it recovers resolved cyclotron motion. Coupled to a compatible field solve, the overall scheme handles variable magnetization (strongly and weakly magnetized particles in the same simulation) and maintains exact discrete charge conservation and exact energy conservation. We demonstrate, through benchmarks in magnetized plasmas, that significant speedups are possible vs. standard fully implicit electrostatic PIC algorithm, while preserving solution quality and the intended conservation/structure properties. [1] G. Chen and L. Chacón, J. Comput. Phys., 487 (2023) [2] O. Koshkarov, L. Chacón, G. Chen, L. Ricketson, in preparation [3] G. Chen, L. Chacón, and D. C. Barnes, J. Comput. Phys., 230 (2011) [4] L. Ricketson and L. Chacón. J. Comput. Phys., 418 (2020) [5] O. Koshkarov, L. Chacón, G. Chen, L. Ricketson, J. Comput. Phys., 459, 111146 (2022)