It is not clear what you are trying to do here. It appears to me that you want to do a meta-analysis. That is, you have results from each site (means, SEs, ...), and now you want to combine to determine an overall mean and SE (effect size mean for the population of sites). If that is what you are doing, MIXED or GLIMMIX can certainly be used, where one overrides the residual with a weight term. There are tricks to the coding that are not obvious. But I don't know if that is what you are trying to do. It is possible that you simply want a hierarchical linear mixed model, but with separate residual for each site, all in one analysis (no meta-analysis). That also can be done, where you get a different residual for each site (sort of like using weights).
I can't give you code until you more fully describe what you are trying to do. Maybe with a toy example.
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