Exporting keychain: You failed to provide the necessary administrator authorization

I had the same, but something was totally messed up and I'm still not sure what caused it.

For one thing, I had somewhere in the process clicked 'always allow' and entered the wrong password. It never asked me for a password afterwards, and kept insisting I failed. Miserably.

Anyway, after I had clicked the lock to 'lock the login keychain' it suddenly started asking for the password again. Then it worked.

I exported two private keys following the steps you reported, and I was able to achieve it both the times.

From the reported error, it seems you entered the wrong password when Keychain Access asked you the login keychain password. I entered the administrator password (the account I used in the test is an account that is allowed to administer my Mac), and I didn't get any error message. By a coincidence, I used the administrative password even when I was asked for a password the first time; I don't think that makes any difference.