I know there are a few homebrewed scenarios with three teams, but has anything official come out that I'm not finding?

Also, has anyone else tried playing with three teams? Any tips for it to run as smoothly as possible?

I have tried and proposed some rules that Miah posted in his blog, I tweaked them a bit having in ming the daedalus rules for new entrances for castle ksiaz and dungeon maps.

General hints, for more than 2p I would rather go for 4 character teams to limit the aoount of figures on the table the maps are intended for 10-12 figures and that way that is preserved, I went for 2heroes and 2 troopers in 2cp games and 3 heroes 1 trooper in 3 cp games to limit the reinforcement options.

The side effect of this is that it shortens turns (and games) which is welcome.

Player order is determined as usual by initiative, only lowest initiative roller can re-roll initiative.

Here they go! (Again it would be good to have some feedback)

Deathmatch:
In multiplayer mode the game ends when there are only 2 factions remaining on the board, the player with more kills wins.
3 players: Each faction team is comprised of 2 Heroes and 2 troopers. On a 2 entrances map use the additional entrances rules on the daedalus rules page 5 (available for free at FFG site). On the unused entry point place a secret door token, any faction may choose that entrance as an alternative entry point. If at end game a player does not have at least 1 kill from each opponent he counts as having 2 less kills.
4 players: Each faction is comprised of 2 Heroes and 1 trooper. On a 2 entrances map use Daedalus additional entrance circles mark ownership of each entrance circle with facedown objective tokens. If at end game if a player does not have at least 1 kill from at least two opponents he counts as having 2 less kills.

Capture the flag:
3 players: Each faction team is comprised of 2 Heroes and 2 troopers. On a 2 entrances map use the additional entrances rules on the daedalus rules page 5 (available for free at FFG site). Each player puts a flag on their entrance circle and the remaining 2 flags in designed objective circles using normal placement rules. The winner must have placed 3 flags and at least 1 from each opponent.
4 players: Each faction team is comprised of 2 Heroes and 2 troopers. On a 2 entrances map use Daedalus additional entrance circles mark ownership of each entrance circle with facedown objective tokens. Each player puts a flag on their entrance circle. Each player puts a second flag on an action circle, if you run out of action circles to put flags on place them on the stamina circle and/or the combat circle. The winner must have placed 3 flags and at least 1 from each non allied opponent, allied faction flags cannot be handled (although allied factions are considered as opponents for all other effects).

Domination:
3 players: Each faction team is comprised of 3 Heroes and 1 trooper. On a 2 entrances map use the additional entrances rules on the daedalus rules page 5 (available for free at FFG site), treat action circles as objective circles for flag placement purposes.
4 players: Each Faction is comprised of 3 Heroes and 1 trooper.On a 2 entrances map use Daedalus additional entrance circles mark ownership of each entrance circle with facedown objective tokens When an action circle has a previous flag already placed use the following rules: If the previous flag(s) belong(s) to an allied faction (Union is allied with Matriarchy and Reich with Shogunate) place your flag on top of it, if there are any previous flags not belonging to your allied faction remove any of those flags from the circle.

King of the Hill Mode
Honestly, I never play this mode so I don't think I can suggest a balanced way of playing this mode.

Objective Mode
3 players: Treat all objective circles as primary objective circles. Take turns as normal placing objective tokens on the circles. Each faction team is comprised of 3 Heroes and 1 trooper. On a 2 entrances map use the additional entrances rules on the daedalus rules page 5 (available for free at FFG site).
4 players: Each faction is comprised of 3 Heroes and 1 Trooper, On a 2 entrances map use Daedalus additional entrance circles. Each faction places 2 objective tokens each, if all objective circles are covered try using action circles, if all of them are covered, place the token in the stamina bonus circle and/or in the combat bonus circle.

Sounds very cool!

I actually plan to play three player some time this week, so I'll give you feedback on how well it works.