I think you are mistaken about what me does, and about the capabilities of custom families. me treats the missing value as a parameter and gives it a likelihood term that is equivalent to giving it a prior distribution corresponding to the measurement error distribution, a procedure sometimes referred to as “Bayesian imputation”.
For your use case, this is just a roundabout way to set a prior on a parameter. This is distinct from multiple imputation, wherein the missing value is repeatedly sampled from the measurement error distribution, as described above:
Stan (for good reasons) enforces that the data cannot change midstream during model fitting. Thus, every quantity in a Stan model must either be fixed permanently (i.e. data, transformed data), or be fitted (i.e. parameters), or be a straightforward function of data and parameters (i.e. transformed parameters, local variables). This means that there is guaranteed not to be a clever solution involving me or a custom family that is capable of fixing the posterior margin in the way that I understand you to be describing. The only options that I’m aware of for fixing the margin are via multiple imputation, via iteratively tweaking the prior, or via reweighting the samples to approximate tweaking the prior (see here New package for sensitivity analysis using importance sampling).
Sorry if I’m still misunderstanding your aim.