Preprint. Structure-preserving kernel methods can recover Hamiltonians from sampled trajectories, but the numerical scheme used during fitting changes the target being learned. The authors quantify this integrator-induced bias for midpoint and symplectic Euler methods, derive finite-sample error bounds that account for observations from the same trajectory, and study robustness to noise. A multistride Richardson correction improves the order of the numerical error without requiring new trajectories. Experiments across several Hamiltonian systems support the theoretical predictions.