Tools for Those Who Summarize the Evidence Base
Hmmm, yours is an interesting problem that probably occurs more often than analysts report. I browsed the Hedges report you linked, and it would rightly seem correcting the statistics would necessitate having the intraclass correlations. Do you have these?
It seems that you are intuiting the difference between the "wrong" analyses and the right ones--no differences in effect sizes but (probably) large differences in the inverse variances. But, again, you need the intraclass correlations to calculate the weights. In their absence, you need to do sensitivity analyses in which you examine a range of possible intraclass correlations and see what impact these have on statistical inference. Does that makes sense?