In a discussion on SEMNET I was made aware that the independence of within- and between level errors is an assumption of multilevel modelling (although I still have a hard time understanding how that assumption could be violated). I was recommended to try and estimate the covariance between level-1 and level-2 errors using Bayesian priors, but I'm not sure this is possible in Mplus?
I don't think it is identified. Errors are assumed independent of predictors to be able to identify the regression coefficient and the between error is like a predictor (random intercept). You can try using Bayesian priors if you want but you will have to convert the two-level model to a wide multivariate model.