Battle of Azanûlbizar

The Conflict

Prelude

Then would come the Battle of Azanûlbizar. Clearly, Moria in this context must be totally taken. No simple killing of orcs would be good enough. This means killing the balrog and the Wraith present. How would this be done? There would have to be two parts. First is the assassination plot. The other is the battlefield.

Deployment

While that is happening, the assassination plan would be going on. The team would involve the most experienced of all the Rangers, hobbits, and elves. Gandalf's dwarven mercenaries would probably be with them too. But most importantly would be the other major players in the White Council. This means Aragorn, Elrond, Galadriel, Radagast and Saruman. We know that Gandalf alone was able to beat Durin's Bane - albeit at the cost of his own life. We also know that the members of the White Council were able to force Sauron himself into a retreat. Even from his fortified stronghold. So, it is not unreasonable to assume that this would be possible. Yet this would not be easy.
And yes, Saruman. We must keep in mind two things with him. First, his fall to the dark side was not an overnight affair. By the time of the Hobbit, he was still loyal to the White Council to enough to participate in this raid. Secondly, Saruman wanted to use the One Ring for his own purposes. He worked with Sauron enough to hopefully get the One Ring first - then turn on him. To me, this means he was never really a true ally of Mordor. Not a good guy either, but no mere puppet of Mordor. This means Saruman would know that if Durin's Bane submits to Sauron, it would become his enemy eventually if all went according to plan. So he would work with the White Council on this mission.
But I think that Saruman would be clearly turning to evil at this point. It is likely that his use of the Palantir would have meant Sauron would be aware of the possibility to recruit him. So the defeated Wraith of the Grey Mountains would be sent to complete the process. The Mount of Sauron and the "Wizard's Ring" meant for him would be delivered as well. So unknown to the White Council, this battle would include a traitor in their ranks whose power is magnified by dark magic. Plus two Ring Wraiths and Sauron's chief lieutenant.

Battlefield

The battlefield would prove to be the simplest and the least important. It would probably also be more of a distraction to bait the Wraith and Balrog into the open. The elves would probably have their biggest contingent present here with the fully formed Dwarf army.

The Engagement

The main army would siege the eastern gates of Moria and try to take everything up to that infamous bridge. This is where we would get to see the full glory of dwarven siege craft at work. Before that though, the orcs would force a battle in the open field. Similar to how the canon version went.
The assassination plan would be to enter from the western gate. It would not be as hard as in the Fellowship, because Elrond would be part of the team. He would remember how to get in. Then they would hunt. Hopefully, the battle would have drawn most of the orcs to the other side by this point. But it would be possible to use eagles to communicate when the battle officially begins. The team could simply wait until that signal is received. Then idea would be to find and kill their targets. This would involve a likely combination of the hobbits doing their thing, traps, fall back positions, the Istari in direct melee combat and elves and rangers providing cover fire. If all else fails, Gandalf would have likely set up plans to lure their targets into the open where the main armies could lend a hand - plus eagles waiting to provide assistance.
Normally if I was a commander, I would want the eagles to come in when the full enemy army is exposed. The application of air power against an exposed enemy that has no counter would make the battle officially one sided. But killing the balrog and the known Wraith are the priority. The idea is that they would need to be exposed. If they knew the eagles were mobilized in the area, they would hide in Moria where they are not in danger. But if they feel they are the ace in the hole, they might sally forth to lend their support to their troops should it turn against them. The result would be them being subject to surprise aerial attack. It is also entirely possible that in desperation, they could be forced out, hoping their remaining troops outside could protect them. Either way, the eagles would come in.

Outcome

This would be important as well in terms of the One Ring. Elrond was there in the Last Alliance. So if Bilbo shows the One Ring, uses it or anything, it might tip him off. Also, Gandalf would not be so stupid. The 77 years it took him to notice in the books seems uncharacteristic of him. Plus, the stress of full combat and the presence of the Ring Wraiths would probably make the corruption take hold faster - thus be more obvious.
The results would be this battle could also involve Saruman's betrayal as he tries to take the One Ring. It could see Gandalf's death and resurrection come from this betrayal at a key moment in battling Durin's Bane. Elrond could recognize what is happening and drive him out. Or it could remain a secret if done right - not exposed until Gandalf is back and healed. I prefer this second option from a simple narrative perspective. It allows for the battles of the Two Towers to take place, while still killing Gandalf in a way that gives him his greater power and defeats the balrog. This still allows Elrond to uncover the One Ring.

Aftermath

This would be one of the more significant battles for the Dwarves. The elimination of the orcs, balrog and wraith all in one swoop liberates Moria from Sauron's influence. Which means that the army of dwarves now mobilized outside has an opportunity to retake it. True, some would prefer to return home. But enough would stay in Moria to see it become a real city again. After this battle, Moria would slowly begin its rise to great power status.
After this, the battle shifts to a whole new area. Mordor was successfully contained. But these were the easy victories. New fronts would have opened up that would be much more difficult. To totally defeat Mordor, an invasion of its territory is a must. But this would mean the defeat of the Easterlings, Southrons and Corsairs. Thus, an attempt at rollback must be done first if there is any hope. This means a piecemeal defeat of these three allies.
Another wrinkle here is the Ring Wraiths. After their eviction from their forward bases during the War of Dwarves and Orcs, alternatives would be needed. Dol Guldor would be perfect for this. So at least one of them would move in. It would begin collecting spiders and orc refugees fleeing dwarf vengeance. So the ground would be prepared for the arrival of the Easterlings. Partially through this accumulating rabble and partly through the renovations they would be performing. They need to get this area ready to host an army in open conflict instead of simply concealing it.

Conflict Type

Battle

Battlefield Type

Land

Location

Moria