Can Android's keystore be backed up?

No. The whole point behind the hardware-backed keystore is that it's resistant to even privileged attacks with root access.

For this reason, not even the OS can access raw key material from the hardware-backed keystore.

While it would be theoretically possible for the OS to do a keystore backup on lower end devices with a software-backed keystore, this is simply not done or allowed by the OS for security reasons.