Skewed binomial data for small p
When fitting a GLM (at least in R), I know there is a optional weight vector that you can include.  This weight is not to give more importance to an observation, but to rather weight observations based on $T$ for example.

The R documentation says:

 For a binomial GLM prior weights are used to give the number of trials
 when the response is the proportion of successes


So I believe this would help adjust for your non-uniformity.  How to choose those weights can be tricky I suppose and can vary a lot with your data, but its worth looking into if it looks like it will help your problem.



Do you have enough data where you can play with dropping data, or maybe if you simulate how much the $SR$ is inflated at low $T$ you can try to adjust for it somehow.  It looks like after about 1000 trials the high success rate issue begins to mellow out, and after 3000 or so you start getting more reliable measures.