Score-based diffusion models have significantly advanced high-dimensional data generation across various domains, by learning a denoising oracle (or score) from datasets. From a Bayesian perspective, they offer a realistic modeling of data priors and facilitate solving inverse problems through posterior sampling. Although many heuristic methods have been developed recently for this purpose, they lack the quantitative guarantees needed in many scientific applications. This work addresses the topic from two perspectives. We first present a hardness result indicating that a generic method leveraging the prior denoising oracle for posterior sampling becomes infeasible as soon as the measurement operator is mildly ill-conditioned. We next develop the tilted transport technique, which leverages the quadratic structure of the log-likelihood in linear inverse problems in combination with the prior denoising oracle to exactly transform the original posterior sampling problem into a new one that is provably easier to sample from. We quantify the conditions under which the boosted posterior is strongly log-concave, highlighting how task difficulty depends on the condition number of the measurement matrix and the signal-to-noise ratio. The resulting general scheme is shown to match the best-known sampling methods for Ising models, and is further validated on high-dimensional Gaussian mixture models.