Hardware keys don't want to transfer the private key out of the hardware gadget, because that compromises the security model. That's their whole point, you can't extract the secrets, the hardware would rather self-destruct than expose the secret. Lack of "sync" (while retaining ease of transferring the key between computers) is the the primary purpose of a hardware security key.

You're opting to a lower security level by wanting something that can be synced.