Classical distributed estimation algorithms for state estimation in Wireless Sensor Networks (WSNs), including consensus-based Kalman filtering and diffusion strategies, typically assume Gaussian observation models under which outliers are rare. However, even a small fraction of outliers can significantly corrupt local updates and propagate errors through the network, degrading the global estimation performance. Furthermore, the distributed setting imposes constraints such as unreliable measurements and limited communication resources, necessitating robust and computationally efficient estimation techniques. This paper proposes a fully distributed estimation framework that integrates a convex, smooth log-cosh loss function within a generalized Bayesian inference formulation to enable outlier-resilient state estimation. The resulting robust update is embedded into a recursive filtering structure and solved via the Exact First Order Algorithm (EXTRA) algorithm for distributed consensus optimization, eliminating the need for dual variables or inner-loop minimization. A formal stability analysis is conducted by conservatively modeling the estimator as a Kalman Filter (KF) with intermittent observations, and a sufficient condition on the robustness parameter is derived to guarantee bounded mean-square error. To improve performance without violating this stability condition, an adaptive strategy is introduced to dynamically adjust the robustness parameter based on the residual magnitude. Theoretical analysis and numerical simulations demonstrate that the proposed method achieves accurate and resilient estimation in the presence of impulsive noise and adversarial disturbances, reducing average Root Mean Square Error (RMSE) by 5-10% under measurement outliers and up to 20% in scenarios with process disturbances, while also requiring fewer consensus iterations for convergence.