No - arrangements can not be classified as classical even if they are intended to be performed in a classical setting.
According to BMI, a song can only be considered classical if it is:
Works that are going straight to recorded release without live concert premieres can not be registered as classical.
Please check with your own Performing Rights Organization to review their specific guidelines for what constitutes as classical. Each society is slightly different, and each has separate processes for performance submission as well.
It should be noted that most works will not fall under the category of "classical". If you have any questions about your works, please feel free to message us in the chat box!