I think maybe some more details about about what you've created in terms of the rainfall might prove helpful.
It should be possible to end up with some rain effect on a transparent background.
The problem likely lies in your workflow. Here is a method for creating rain where you can achieve transparency:
However even If your rainfall is on a black background you should be able to composite it over your footage with the correct blend mode so the black is multiplied out but ultimately without any details about how you created your effect it's hard guess what the solution is.
There is composite with original check box that turns on and off transparency. CC Rainfall actually uses the background image to simulate the way light would wrap around raindrops so you'll loose that enhancement if you just export and lay the light streaks over another image. You just end up with white particles on top of your other footage.
You can make that work better if you use a blend mode like lighten, screen or add and adjust the opacity of the rainfall layer, but if you are going to use blend modes to do the composite, which is the recommend procedure, you do not need to render with an alpha, just add CC Rainfall to a black solid, render, then import into Premiere Pro and try using blend modes and opacity to complete your composite.