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). Those ‘detrended’ residuals are what @ColinKG is getting from spdep.
When you remove autocorrelation from the residuals, then the Moran coefficient becomes a small negative number (like -0.05).