Simulation-based inference for multivariate stochastic differential equations is limited by two interacting difficulties: stochastic forward simulations are often poorly aligned with observations, and numerical solvers may be unstable or inaccurate across parameter regions explored during inference. We address both through approximate Bayesian computation with sequential Monte Carlo (ABC--SMC). First, we introduce an exponential tilting of the simulator-induced summary law toward the observed summary, with automatic calibration of its strength. The tilted law uniquely minimizes a scaled expected squared discrepancy from the observed summary plus the Kullback--Leibler divergence from the simulator-induced law. We show that its normalizing constant is exactly a Rayleigh-weighted average of ABC likelihoods over tolerances and use it to construct a lookahead proposal favoring parameters with high average ABC compatibility across tolerances. Second, we identify a class of conditionally affine square-root diffusions and develop structure-preserving splitting methods. We characterize when coordinate-wise splitting introduces a spurious drift and provide an alternative that applies the diffusion jointly. Across biochemical reaction networks and a stochastic Hodgkin--Huxley model, the splitting methods stabilize inference at substantially coarser time steps than Euler--Maruyama, while tilted ABC--SMC reaches comparable posterior accuracy with approximately three to seven times fewer simulator calls.