Selling the private key would be a bad idea since you can't guarantee the seller destroys it. But you could have the staker address be a contract, with an ownership that can be transfered.

You could use an HSM to hold the private key and then sell the HSM. Overall it's still a bad idea to buy an HSM because you don't know what other transactions have been signed, but maybe the HSM keeps a history of everything it has signed as well.

Which, if you can still sell your staked coins, what is the point of locking them up in the first place?

Good point. Maybe they'll only allow deposits from non-contract addresses.