When you extend a class, you always extend a script, not a scene! That is, you cannot assume the existence of any child nodes – unless you create and add them in the script itself.
Since the number of different chess pieces is rather small, I’d probably create one base scene establishing the common fundamentals and then create an inherited scene for each different piece. So instead of re-using the same Piecescene with a different script, I’d look at the FEN string and then spawn the appropriate piece right away.