I think the CMC doesn't say that a mind can never be right without being causally entangled with the system it's trying to be right about; just that if it is right, it's down to pure chance.

No, the CMC says that if you conditionalize on all of the direct causes of some variable A in some set of variables, then A will be probabilistically independent of all other variables in that set except its effects. This rules out chance correlation. If there were some other variable in the set that just happened to be correlated with A without any causal explanation, then conditionalizing on A's direct causes would not in general eliminate this correlation.