Even if we accept your premise, there are plenty of factions that have no claim to being a "clear choice," like Daibei and Norris. Neither of them deny Lucan's right to the throne, they just go their own way and the military follows them. Additionally, those who express doubt as to Lucan's claim like Margaret and Bzrk, still have no right of their own but their regional military forces follow along too.
Only Dulinor's breakaway faction has an actual "claim" to the throne, and he killed the emperor. If I were the emperor, I would make personal loyalty to me a focus of any military training and indoctrination, so that at least a sizable minority would have problems following my killer over my heir.
Norris actually DOES deny Lucan's claim - by virtue of not having been confirmed by the Moot, Lucan is still Prince, not Emperor. Norris also claims the title "Regent" and denies wanting the throne for himself. He's also got TWO hostile "rebel" domains between him and Capitol, so it is kind of moot. (pun intended.)
Let's review the claims -
Lucan - By Right of Inheritance. Ignores requirement for moot confirmation, and as acting emperor, bars the moot from meeting to confirm him (or more likely, refuse to do so and appoint someone else as Emperor). Shown as a ruthless sociopath.
Dulinor - By Right of Assassination. Doesn't hold Capitol, doesn't quite meet up to the prior standards of the right of assassination. Archduke of Illellish. Had broad internal support for taking the throne within Illellish, and extant obedience as Archduke from most of the Dukes and barons of Illellish.
Margaret - By right of inheritence, with following caveats: Lucan's inelligible for his high crime of Murder (of Varian), Dulinor didn't hold the throne, so is just a criminal. Requires the moot to first reject Lucan and Dulinor to be a valid choice.
Domain of Sol - No claim, but fighting a local war and hoping to hell not to get drawn into the civil war as well. Pretty well ignored.
Craig of Diabei - There is no clear emperor (until the moot elects one). Until then, my best bet is to sit it out. And, no, Illellish won't go after Dulinor.
Brzk: I want to be Emperor, but since it's clear no one else considers me fit, I'll preserve antares as an independent state.
Real Strephon: By right of having been confirmed by the Moot. Dulinor got the wrong man. Lucan's gone off the deep end. Not believed by many. (Establihsed in canon to have actually been the real Strephon)
Vland: The Vilani Cultural Region simply reverted to being the 1st Imperium - the constituent megacorps were still around, and in their cultural region, still running things. For the locals, no change, other than the ouster of the local Archduke from power, and new enemies on 3 sides, plus a neutral and an old enemy.