No, you can get the clone at the very beginning of the game. It will sit in the house until you need it for the Chrono Trigger quest. It'll be in your room, right by the window, and if you go up to it and talk to it, it switches to several different poses.

Anyway, I don't remember the SP amounts, but if you did the game with the movement matching, and you won, you should have gotten it. After you win it, you just get cat food. Are you sure you won?