I believe the keyboard part was added to the mono mixdown directly, so doesn't exit on the multitracks used for the Hawthorne remix. We have the multitrack tape out on boot (SOT) and there is no keyboard part.
I seem to recall a bootleg of the unedited mono mixdown tape - Brian's adding the keyboard while a female voice (Marilyn, presumably) dubs over a vocal. Brian keeps flubbing the keyboard, and we hear the track being rewound. As they had sel-sync long before this point in time, I presume you're right in that the keyboard was recorded directly to the mono mix while it was being mixed.