After some thinking, I'd say the campaign mode should be a pure coop thing, being most similar to the team castle sweep mode of KA, but with a couple of fixed objectives on maps (and maybe a limitation on what chars can play, though that could be a setting before starting a map, i.e. suitable characters only or any character). In an extreme situation this could simply include e.g. all 9 maps of Heretic/City of the Damned with very little tweaks, just updated for KA coop instead of Heretic coop.
Feel free to discuss or bring new ideas, the campaign mode is not for 1.0 but for 1.1.