The Black Riders Rulesheet

If a hero is in a player’s discard pile at the end of the game, that hero’s name is added to the list of Fallen Heroes in the Campaign Log.

What constitutes the "end of the game" for the campaign? In a 2-player game, we lost Bilbo [Hero] on turn four due to an unfortunate Shadow Card, and over the next few turns we lost the remaining heroes (our momentum was lost). Since we lost the game, does this qualify for "end[ing] the game"? Does this mean that, when we retry this mission, we cannot use Bilbo [Hero] and take the +1 threat due to the rule at the beginning of this post?

You're free to retry a scenario if you don't like the result... I believe they put that in the rules insert for either Black Riders or The Road Darkens (which contains a campaign FAQ). You only have to put a hero on the Fallen Heroes list if you win the scenario and decide to accept that result and go on to the next scenario.