I had a similar issue when I was running my lights. I dried out some connections and sealed them and the GFI stopped tripping.
To expand on caretaker, the exact current difference is 5 milliamps between the hot and neutral. That is very small. So it doesn't take much leakage current to get into some water and trip your breaker/receptacle.
Have you tried plugging the circuit into a non-GFI receptacle and see what happens. If it runs, then you have something faulting to ground and since you said it happens when it rains, that is classic ground fault. Somethings getting wet or something is getting moisture in it which is causing the problem.
If you think it is the transformer, then put it on some rubber or something to keep it from grounding itself that way you can track down the problem. If the problem is inside the transformer and the case is grounded, that might be where the problem is. You will just have to narrow it down.