Cannot get simple residual autocorrelation tests to work

[cmcd]

If I follow you correctly then this one may answer part of your question (though without code):

Correcting spatial autocorrelation using SAR - different results with brms vs spatialreg packages?

It looks like that method brms:::posterior_predict_gaussian_lagsar is taking draws from the SAR model, generically: y \sim Gau(\mu, \Sigma) with the SAR specification of the covariance matrix. That will generate spatial autocorrelation (SA) in the residuals, y - \mu (as it should). The model implicitly has a spatial trend. What spdep is doing is calculating the implicit spatial trend embedded in the covariance matrix, then it calculates residuals as basically y - mu - SA (just the general idea)…

depending on what your project requires geostan may offer something for you