3D brain MRI studies often examine subtle morphometric differences between cohorts that are hard to detect visually. Given the high cost of MRI acquisition, these studies could greatly benefit from image syntheses, particularly counterfactual image generation, as seen in other domains, such as computer vision. However, counterfactual models struggle to produce anatomically plausible MRIs due to the lack of explicit inductive biases to preserve fine-grained anatomical details. This shortcoming arises from the training of the models aiming to optimize for the overall appearance of the images (e.g., via cross-entropy) rather than preserving subtle, yet medically relevant, local variations across subjects. To preserve subtle variations, we propose to explicitly integrate anatomical constraints on a voxel-level as prior into a generative diffusion framework. Called Probabilistic Causal Graph Model (PCGM), the approach captures anatomical constraints via a probabilistic graph module and translates those constraints into spatial binary masks of regions where subtle variations occur. The masks (encoded by a 3D extension of ControlNet) constrain a novel counterfactual denoising UNet, whose encodings are then transferred into high-quality brain MRIs via our 3D diffusion decoder. Extensive experiments on multiple datasets demonstrate that PCGM generates structural brain MRIs of higher quality than several baseline approaches. Furthermore, we show for the first time that brain measurements extracted from counterfactuals (generated by PCGM) replicate the subtle effects of a disease on cortical brain regions previously reported in the neuroscience literature. This achievement is an important milestone in the use of synthetic MRIs in studies investigating subtle morphological differences.