I think one of the main reasons the story mode campaign goes untouched by many players is that it is completely detached from the multiplayer portion of the game. I love the story personally, but I feel it is truly limited in scope by the fact that you HAVE to play particular characters for each mission and you gain absolutely NO EXP for characters in the multiplayer spectrum. There are plenty of other games where you can play alternative characters through the storymode, the cutscenes just keep the original intended character and then the map loads up with the selected character instead. I feel it would give more incentive to play through the storyline if people could play through the game with the character they want to play, and simultaneously earn a little bit of experience to level up their characters for multiplayer. You can still keep the story-driven feats exactly as they are, simply allow the player to choose their hero and transfer exp gains from the campaign over to the faction war.
I agree that something like this would be good, it would be a whole lot of fun to kill Apollyon as a Lawbringer, and adding the experience to multiplayer would be great.
To expand on it you could also add in the XP multiplier for difficulty, so completing at realistic will boost xp gain.
And for times when your internet is playing up and you can't connect, this is perfect for that.