I'm guessing the poison is either
- something that the prince is allergic to
- something that only becomes poisonous when mixed with something else, like the prince's favorite coffee or something
- something that is only poisonous in high doses. The poison tester only eats one cookie and Lily can simply excuse herself saying she's full, forcing the prince to eat the rest on his own
Honestly, those are my exact thoughts as well.
My mom isn't allergic to dairies, but she just can't eat cheese without getting sick (I inherited half of that trait from her and get to the point of almost throwing up with the taste of cheese, but I can eat it normally if I disguise the taste with something else of stronger flavor), so my first instinct here would be to suspect the prince is allergic to one of the ingredients she uses in the cookies.
But that would be too simple for a manga, so my over analytical mind that I developed after reading way too many Conan murder cases veers me to the idea that she is something making poison in front of him without his knowledge, like mixing specific herbs on his drinks in small doses so that he becomes weak over time, but since no one has any knowledge of plants or herbs for medical purposes, they only see it as flavoring or something beneficial.
Another theory I came up with is that she somehow made the food tasters immune to the poison, but not the prince. How she would do that, though, I have no idea.
I mean, if it were the first one, why would the villainess go through all this trouble instead of just stating the prince is allergic to the food she's giving him, right?