In Bayesian causal inference for partially identified parameters, there is a delicate balance between parameterizing models in terms of the fully identified and unidentified parameters directly versus modeling the parameters of primary scientific interest. We explore the challenges of Bayesian inference for partially identified models in the context of multi-treatment causal inference with unobserved confounding in the linear model, where the treatment effects are partially identified. We demonstrate how carefully chosen priors can be used to incorporate additional scientific assumptions which further constrain the set of causal conclusions, and describe how our approach can be used assess robustness and sensitivity of the outcomes. We illustrate our approach to multi-treatment causal inference in an example quantifying the effect of gene expression levels on mouse obesity.