As of yesterday, Koopa is officially not scared of the toilet in his daycare anymore! I picked him up and he explained proudly (but silently, because he doesn't talk there) that he used the bathroom.
It probably helped that he felt sick a few days ago and had to throw up into that toilet. And once you develop a throw-up relationship with something, you know that the two of you can get through anything together. We're all very proud of him. Now he just needs to conquer all those other toilets out there in the world. One toilet at a time.