DSFinV-K - Do I need to create a Cash Register via the DSFinV-K API even though I have already created a client via the Sign DE API?

Yes.
Via the SIGN DE API only the client_id and the serial_number of the POS device are submitted. In the DSFinV-K, however, additional information about the POS device is needed (brand, software, ...).
Therefore, a cash register must also be created via the DSFinV-K API with the same client_id that was used in the SIGN DE.