Is it possible to derive a new private key from an existing private key and retain the same bitcoin balance without any transactions?
No. A private key corresponds to one public key (and vice versa), and that corresponds to one address (and vice versa). Having a new private key (regardless of where it comes from, so it could be derived from a different private key) would mean that you have a new public key and a new address. Thus you cannot spend any funds associated with the first address with the new private key because it is not associated with that address.