ATM I have all live strokes and have tried clipping masks, pathfinder, shape builder, and am not able to get the same effect. Here’s the "live version I’ve done:
I think you do, but maybe not in the way you think. Copy the pink-cloud stroke, paste it back in the same spot and increase it’s width by two or three times. Expand the stroke to a shape, then, while selecting this shape and the rainbow strokes/shapes, use the pathfinder tool to cut them apart. Then just select and delete the leftover parts.
You can also copy and outline the path/stroke of the cloud shape. Then you can ‘offset path’ (not much, maybe a 1/6th to an 1/8th of an inch). Use that new shape with the pathfinder to cut away the rainbow.
Thanks! That solution worked. I swear I had tried that multiple times but wasn’t getting the right result. I did have to make the rainbow strokes into shapes for it to work. So props to @Carlo too.