Taking into account sampling error in MRP

Imagine you have estimated a MRP (multiple regression and post stratification) model and want to get population estimates, but you don’t have census data. Instead, you have population estimates from a survey with known sampling error. How do you incorporate that sampling error into the predictions made through the MRP model?

Draw from a (presumably) normal distribution with the population mean and standard error S times (if you have S draws from the posterior distribution) and poststratify to those realizations.