This paper establishes a noncommutative twisted version of the GrothendieckRiemann-Roch (GRR) theorem on rank-2 polarized K3 stacks, reconstructing the Euler characteristic of divisor classes as alternating sums of Ext groups through skeletal filtration constructions in derived categories. Based on the 23 cohomological jump trajectories observed on Picard lattice U(2)-type K3 surfaces in previous work, this paper proves that these phenomena correspond to nontrivial extensions of canonical skeletal filtrations of twisted sheaf complexes. The core contribution lies in providing the explicit formula χ(Oα X(D)) = ∑2 n=0(−1)n dimExtn Db(X,α)(En,Fn), where {En} is a sequence of rigid sheaves determined by the quasi-wall structure. Theoretical results are numerically verified on quartic surface stacks using Macaulay2, covering effective divisor classes in the parameter range |a|,|b| ≤ 15, consistent with predictions.