Accurate characterization of subsurface flow is critical for Carbon Capture and Storage (CCS) but remains challenged by the ill-posed nature of inverse problems with sparse observations. We present Fun-DDPS, a generative framework that combines function-space diffusion models with differentiable neural operator surrogates for both forward and inverse modeling. Our approach learns a prior distribution over geological parameters (geomodel) using a single-channel diffusion model, then leverages a Local Neural Operator (LNO) surrogate to provide physics-consistent guidance for cross-field conditioning on the dynamics field. This decoupling allows the diffusion prior to robustly recover missing information in parameter space, while the surrogate provides efficient gradient-based guidance for data assimilation. We demonstrate Fun-DDPS on synthetic CCS modeling datasets, achieving two key results: (1) For forward modeling with only 25% observations, Fun-DDPS achieves 7.7% relative error compared to 86.9% for standard surrogates (an 11x improvement), proving its capability to handle extreme data sparsity where deterministic methods fail. (2) We provide the first rigorous validation of diffusion-based inverse solvers against asymptotically exact Rejection Sampling (RS) posteriors. Both Fun-DDPS and the joint-state baseline (Fun-DPS) achieve Jensen-Shannon divergence less than 0.06 against the ground truth. Crucially, Fun-DDPS produces physically consistent realizations free from the high-frequency artifacts observed in joint-state baselines, achieving this with 4x improved sample efficiency compared to rejection sampling.