It has been my experience that the GFCI is bad. I would replace it with a new bubble cover and 90% of the time problem solved.
Is the transformer fully loaded? Usually I don't have in rush issues unless it is close to max. I have had issues if the line between the GFCI and the breaker is short, less than 20' or so.
If you have a bad LED, or a short, it would show up in your amp probe reading of the home runs.
Tommy Herren, CLVLT #1169