Yes they fit. I recommend putting the bottom in first and you'll have to squeeze the top of the bucket a little and it fits with no problem. We do it all the time. Makes it easier to get out too (except Velocicoaster, but still isnt an issue). Definitely should be empty or mostly empty. The lid will not fit while on it obviously. .
I was just coming here to ask the same question. Seems like a bad idea that the didn’t just adjust the dimensions of the bucket slightly to better fit as it sounds like it’s a bit of a squish to make it fit.