"Turns out, you can't play the same game with different Steam accounts." I don't know if you actually meant to say it this way, but it just sounds like multiplayer games cannot exist the way they do right now.
Or just use the same Steam account in both computers, use offline mode Steam in one and online/offline in the other.
Does it work though? You can't play a game on someone's Steam account if it already runs on a different PC in the same account.