In Chrono Cross, you can ascribe all the preplanning to Belthasar. I'm assuming that Belthasar had some contact with her, or perhaps planned this through Lucca.
My main problem with Kid is her Chronopolis dialogue. She suddenly demonstrates a bunch of weird knowledge about all sorts of things. My only explanation is that it came as a result of her brainwashing by Lynx, though this doesn't make sense. However, if you say that Lucca was the one who set Kid up to do certain things as Belthasar's arm, then perhaps she did have this esoteric knowledge of the Flame and other things all the time.
It just seems unlikely given her demeanor and behavior for the entire game prior...