Sodium-ion batteries are promising for cost-sensitive and low-temperature electric-vehicle applications, yet reliable battery management systems (BMSs) require evaluation beyond a development cell. This study develops a lightweight multi-task gated recurrent unit (GRU) framework that integrates timestep-level state-of-charge (SOC) estimation with window-based, cycle-informed state-of-health (SOH) estimation. The main A6 configuration uses aging-sensitive descriptors and contains 24,882 trainable parameters. On the NFM source-domain test set, A6 achieves 1.078% SOC root-mean-square error (RMSE), 0.978% cycle-aggregated SOH RMSE, and an SOH coefficient of determination (R²) of 0.835, compared with 2.409% SOH RMSE for the constant-mean baseline. An ablation analysis distinguishes the effects of task coupling and input observability. A separate three-input source-trained backbone is evaluated for external SOC transfer: on Hina, zero-shot RMSE is 22.5–36.2%, and calibration with 25% of the target cell’s 25 °C records reduces errors at unseen temperatures to 14.0–27.4%; on the CVUT 4S11P cold-start battery pack, zero-shot RMSE of approximately 27–29% decreases to 12.0–19.7% following adaptation on the first 25% of time-ordered records. The separate source-domain and transfer tests show why low in-domain error alone cannot establish SOC accuracy on a different commercial cell or battery pack, even when limited target-domain calibration is available.