Invasion of Minas Morgul

For the moment, lets assume that the One Ring is not destroyed and some form of attack is needed as a distraction.

The Conflict

Deployment

Honestly, I am a bit unsure of is what to do with Faramir. As he is the rallying point for possible coup forces, sending him to lead this attack on Cirith Ungol would keep him out of the way and not causing trouble. Sending him down to rally the nobles would have the side effect of possibly putting him in contact with enemies and at greater risk of defection. But, if he remains loyal, there is a higher chance that his personal friendship with a lot of these people would make them easier to persuade into loyalty as well.
To balance this, I would make sure Faramir knows that Eowyn is to stay in Minas Tirith as an implied hostage. Then the second in command for his group would be ordered to spy on him and kill him if he defects. After taking Cirith Ungol, Faramir is to be sent to the most powerful lord (Imrahil of Dol Amorth with 12,000 me available). This territory is 350 miles from Minas Tirith.
For Cirith Ungol, all one needs to do is send in 100 rangers and 50 elves to man the walls. I would also send up 100 Tower Guards and 100 dwarves to handle defense from direct assaults from Mordor's forces. I suspect that the forces devoted to Minas Morgul would include the 16,800 Compagnie d'ordonnance and 45,000 Oathbreakers led by Aragorn personally. The absolute need to take this city and the need for Aragorn to be seen as the one doing it would justify this deployment to me.
An important note here on the Dwarves. The recent racial violence in Minas Tirith would have seriously inflamed tensions. People would not have forgiven the Dwarves for their actions by this point. The siege itself would probably make things worse. The Dwaves would not just be be blamed for the Southron/corsair devastation of the south. The risk of Minas Tirith falling because of the ill timed destruction they caused in the city would be an issue too. The violence against Dwarves would probably only escalate. So just as Aragorn needs to be seen as leading this battle himself, he must also be seen as excluding the Dwarves from such important symbolic acts.

Battlefield

First, I think my proposed attack on Minas Morgul would fill the need for a diversion quite well with the numbers available. Cirith Ungol is a ready built fortress that would be vulnerable to surprise attacks at this point. The fortifications and narrow pass is also primed for defense by a small number of highly skilled troops. Add in the fact that this pass ends with the also well fortified Minas Morgul.
Then there is are major strategic and symbolic reasons for taking the city. Like the Black Gate, it grants access to the Plateau of Gorgoroth. With that comes access to Barad-dur and the main industrial power of Mordor. Plus Mt. Doom with all the importance that entails. What little information Gondor would have suggests that the Ringbearer passed through here on their way to Mt. Doom. So any attempt to send last minute aid would benefit from this route.
Then there is the value of Minas Morgul itself. Any exploitation of the manufacturing power of Plateau of Gorgoroth or the agricultural riches of Nurn would benefit from control over the Morgul Vale. Through Minas Morgul, one can export all of Mordor's economic output. This then allows it go get to Osgiliath and thus to the rest of Middle Earth. Then, taking here would essentially block any resurgent Mordor army from coming through here for a second attack on Gondor.
Lastly, the long occupation by Sauron's forces means that huge symbolic value would be invested in the place. By retaking the city, Aragorn can signal the final defeat of Sauron. He can also demonstrate that victory at Minas Tirith was not just a fluke. He can do this in a way that overcomes one of the more embarrassing moments of Gondor's history. In the process, Aragorn will be able to play on the revanchist/irredentist feelings seeking to return Gondor to its former glory.

The Engagement

Ideally taking Cirith Ungol would provoke a response. The goal would be to use this response to kill as many orcs and trolls as possible. Due to the location and design of the fort, I think these numbers could hold for a long time. The tunnels in the area could allow the same retreat and don’t die methods used to survive Minas Tirith. It must be noted here that Shelob would be dead by this point. Also, due to the approach from the back, The Watchers would not block access either. The garrison was also dead due to infighting. So occupation of the fort would be quite easy.
Once these forces are in place, the assault on Minas Morgul would begin. This too would be relatively easy. Especially when the local forces would already be engaged in battle. Once the city is taken a pincer move can be used to encircle the remaining orcs.

Outcome

Many important possibilities would come from this action. First, the ability to get aid to the Fellowship becomes possible. For this purpose, I would bring up the Grey Company. Their objective would be to find the Fellowship and get it to Mt. Doom.
Next. Barad-dur becomes exposed. This would be another critical symbolic action. So I would make sure a garrison is left to defend Minas Morgul - 3,000 should be good enough at this point. The Army of the Dead and at least 10,000 of the Compagnie d'ordonnance should be taken up to claim that tower. By this time I fully expect the Black Gate to be seized. This allows a pincer move to eliminate the remains of Mordor's army.
Just as important to me is the One Ring would most likely have been destroyed. The result is the destruction of Barad-dur, the Ring Wraiths that remain and all cohesion in Mordor's army. Which means that there would be no siege. Effectively, the area would be occupied after marching up. A few small skirmishes might happen with small and disorganized orc bands. But that would be it.
Immensely important from both a propaganda and economic perspective is Nurn. Should Aragorn send the 3,000 Compagnie d'ordonnance left and the Rangers from Cirith Ungol, this area could probably be liberated. This would be doable mainly because there is little military force in the area to resist - even before the defeats crippling Mordor's forces. What would be there is essentially just the overseers controlling the slaves growing the food in the area.
And there is the value of sending the force to take the area. The last hope of Mordor winning is lost with the ability to produce food for its armies. This alone would mean game over if Gondor was able to sustain its own war efforts long enough. But just as important is the slaves themselves. Once Sauron is overthrown, these slaves would be liberated regardless of what happens. But by sending this force down, their liberation can be seen as an intentional act by Gondor. Their liberation via intentional policy instead of freedom being an accidental byproduct of other policy would go a long way to helping Aragorn retain power.

Conflict Result

Defeat of Mordor