Update: Effect sizes are in fact independent of one another, and not correlated as I thought. That should make things easier.

Approaches that may work:

1) Using bootstrap methods to create confidence intervals. Basically, resampling so as to generate a list of several thousand slopes, and then estimating error from there. Guidance would be appreciated, as I am not experienced with this.

2) Performing linear regression using the most conservative estimates, by plotting values (Effect Size) that have the error added (or subtracted) to them. This would overestimate the error, however.