Flexible coupler is a promising solution for enhancing wireless network capacity by moving passive couplers around a fixed-position active antenna to reshape the induced currents on passive elements. Motivated by this, this paper proposes a novel flexible coupler array that incorporates additional degrees of freedom (DoF) in radiation pattern reconfiguration and enhanced communication coverage with low hardware cost. Specifically, a new form of mechanical beamforming can be obtained by moving only the passive coupling elements while keeping the active antenna stationary. In addition, the flexible coupler antenna can slide along a rail toward users, thereby enhancing communication coverage. To fully exploit the potential of the flexible coupler array, we formulate a two-timescale sum-rate maximization problem with statistical channel state information (CSI). The antenna position is optimized based on scattering cluster-core statistics in the slow timescale, while mechanical beamforming is optimized based on multipath channel statistics in the fast timescale, subject to movement and energy constraints. To address the coupling between timescales and the high cost of extensive channel sampling, we develop a digital agent framework that leverages an electromagnetic (EM) map to generate statistical channel information for different user and antenna positions. Then, a deep neural network is trained to learn a slow-fast performance (SFP) surrogate. Mechanical beamforming at the fast timescale is obtained by selecting per-antenna radiation patterns from a predefined dictionary via a convex relaxation. Simulation results verify the performance gains achieved by the proposed flexible coupler array and the digital-agent-assisted algorithm.