One thing I’m pretty sure of is that by this time the army would no longer have been organized by tribes. At least not with tribal heads leading them. One of the key changes introduced from the time of Saul is the move toward a United Kingdom, which meant subsuming tribal identities. That, I believe, is one of the subtexts going on in the struggle between the Omirides and the rural people Elijah represents. This unification would never be total, but by Omri’s time, it was – I think – complete enough that when you fought, you fought as part of the Army of Ephriam and not for the honor of Gad or Dan.
Not to say they didn’t organize around “the clan of Dan” or some such in their battle order. They may have. Compare them in this regard to the Scottish clans fighting as Scots against the hated English. Dual loyalty, but a movement toward a national identity first.