High-fidelity electromagnetic-transient (EMT) simulation is indispensable for evaluating grid-forming (GFM) converters, high-voltage direct-current (HVDC) interactions, current limiting, and protection behavior. However, exhaustive evaluation across device locations, ratings, operating conditions, and contingency combinations is computationally burdensome. This paper develops an operator-supervised digital-twin screening framework that prioritizes GFM STATCOM and energy-storage-enhanced STATCOM (E-STATCOM) studies before detailed EMT validation. The framework is anchored to the 2020 eastern Denmark (DK2) disturbance, where sequential disconnection of the BJS–ISH and BJS–HVE 400-kV corridors weakened the BJS/HKS area, preceding the BJS synchronous-condenser trip and Storebælt HVDC blocking. A public-data representation containing 127 buses, 426 directed message-passing edges, 23 node features, and 10 edge features was combined with 3,319 device and operating scenarios. A six-layer topology-adaptive graph model with conditionally applied physics penalties achieved an area under the receiver-operating-characteristic curve of 0.9618 and an average precision of 0.9445 against paper-derived weak labels. At a 0.5 decision threshold, unstable-class precision was 0.9248 and recall was 0.6675. A Brier score of 0.1413 and expected calibration error of 0.1530 indicate that the model outputs should be interpreted as ranking scores rather than calibrated physical probabilities. Reduced positive-sequence ANDES studies indicated a 97.37%–97.49% reduction in the scripted maximum frequency deviation for the tested converter-support cases while retaining the represented 600-MW Storebælt transfer. ONNX Runtime CPU inference required 5.805 ms on average and approximately 19.3 s for the complete scenario archive. The evidence supports rapid engineering prioritization rather than EMT replacement, autonomous control, or independent verification of DK2 physical stability.