Hmm. This would need a fairly substantial redesign in FIDO and might make the hardware more expensive, but in outline it sounds viable.

The hardware today doesn't have a "base private key" (or public key) in the sense you mean. I feel like you're thinking about this like it's RSA, which it really isn't. RSA is weird because Encrypt and Sign are related operations, that is not how most things work. So you'd have to use key agreement plus symmetric crypto, and you'll struggle to fit everything into the current data structure sizes.