OK. I think I see where you are going.
/spitball mode on
What would happen if you ran proc mixed as you did before, got the residual error, then plugged back in to fit each individual lake separately, but with the residual parameter fixed at the shared value.
/heresy mode on
Or took the value as an informative prior on the residual error, and used the BAYES option in PROC GENMOD, with lake ID as a by variable. You would need some estimate of the variability of the residual error, but you could get that from PROC MIXED and calculate what you need. I think there is a pretty good worked example in the PROC GENMOD documentation. ODS output the posterior estimates, and then you could do all of the positive, negative, no change calculations needed.
By using the shared value as an informative prior, you might get around the memory problem of trying to do this all at once. Might take a full day to run, and the diagnostics can get pretty hairy and about thirty other things, but in my opinion it's an interesting approach to the shared variance idea. Plus you can see how much it moves from the common value for each ID--that might be another interesting classification exercise.
SteveDenham