Southern Conquests

The Conflict

Deployment

So Pelargir is likely to become the center point for operations against the Corsairs. The garrison there that resisted the attacks before Minas Tirith should be left there. Their orders should be to sally forth to attack any Southron army it saw coming that it could defeat in the field. Assuming of course that said army was heading north instead of towards Pelargir. If it was heading to Pelargir this army should stand back and accept the siege.
This means that the forces in the area would probably be roughly 500. I'm setting that low to account for the losses taken during the battles earlier. The 11,200 men left defending the Southern Fiefs would be needed to reinforce here. But with the stunning victories in Minas Tirith and Pelargir, it would be easy to convince the Southern Lords to send the rest of their forces to eliminate the threat for good.
Also important is the fact Gondor effectively has a navy again. The Corsair ships used to move troops to relieve Minas Tirith would be moved back Pelargir. To me, they would need a bit of reworking to incorporate them into Gondor's military. For example, the logos and so forth would have to be repainted to reflect their new owners. More important would be the redesign of their interior. Gondor's need for traditional troop transports would not be well served by ships meant for capturing loot.
A note on a few of my choices of priorities here. I left 1,000 Cohorts with Pelargir not because it was less important but because it was closer to the more professional troops coming in from the rest of Gondor. Thus it needs less initial protection. I also don’t think it would have been as devastated by the war. The fighting is likely to have done some damage, but this is not going to be on the same scale as the area around Minas Tirith. Which means that it is more intact – with more residents, more functioning infrastructure and so forth.

Battlefield

The first thing of note here is the geography of Corsair territory. Umbar would be a highly valuable port and is the base of the Corsair threat. Part of this economic value is the river. By taking Umbar, one now has access to all the plantations upstream. Taking these would deny the enemy vital economic assets while gaining control for Gondor at the same time. From this same river base, the war against the Haradrim can be more effectively waged.

The Engagement

To me the first step to deal with the southern theater would be the taking of Umbar. Once the ships are ready, the Pelargir force would sail out and take Umbar. Then they would move up the river. The plantations would be seized, local owners and overseers would be deposed. These would probably be used as part of the compensation for soldiers. The local slaves would probably be freed.
From there, it is a matter of dealing with the Southrons. This campaign is going to be much different than the Easterlings. First, the Southrons are a group of nomadic tribes rather than a more unified state. They also rely more on archery and mounted shock attacks instead of massed heavy infantry. These are going to be important.
The first thing to note is this is likely to be a very long campaign. Also, there is likely to be many internal divisions and rivalries between tribes. Historically, there are always weaker powers willing to ally with a strong outside power to gain support in their wars against rivals. Great powers are always willing to accept this friendship as the alliance provides a foothold in an area they might have designs on. The British Empire mastered this tactic. The problem is that the minor power is able to defeat their rivals in the short term at the cost of becoming the collaborators to the imperial project of their sponsors. This is what I expect to happen here. In exchange for concessions, Gondor will support the minor tribes in the Harad and grant administration posts to influential tribal decision makers. The end result is a long running series of inter tribal wars intensified and sponsored by Gondor.
Gondor’s forces then provide brute force to move into the chaos to mop up and occupy the best areas. This would allow for Gondor to stretch out the operation. A full fledged conquest and occupation would take more troops and money than can be sustained by Gondor after the brutality of the War of the Ring. But a more piecemeal approach like this is much more manageable.
Just as important are the elements of Gondor's doctrine and mobilization. The main source of troops would be the Compagnie d'ordonnance. This is a very short term mobilization not conducive to long campaigns. It does however complement the bite and hold idea Gondor already employs. Each campaign season would see small gains made in key areas. Then these regions would be consolidated over the next few months as Gondor waits for the next campaign season. This would set up defensive positions preventing enemy counterattacks while integrating the area into Gondor. The next offensives would then have a base of operation closer to the next target.

Outcome

This conquest would go a long way to securing the southern borders of Gondor. The most obvious is the defeat of the main naval threat to Gondor. This would also eliminate the threats to the economic development of the Southern Fiefs. Then, the resettlement of troops in the area would create a defensive buffer. Their involvement in the economic activity in the area would then further economic integration of the area. There is the fact all of this is centered on a river - a natural defensive barrier and highway at the same time.

Aftermath

The first thing would be to begin resettlement of the mobilized troops. This could easily started from a foundation of the Corsair plantations. Though this poses a bit of an issue. Emancipation is completely justified in moral terms and as a means to break the former rulers of the region. But it would also deny the new owners of the plantations labor. After the US Civil War, a somewhat similar situation arose. The owners had land and equipment but no means to pay workers. The workers could no longer be forced to work for free. But they also had no land. The solution turned to was sharecropping.
A similar set up could be used for the Corsair plantations. The new Gondorian settlers would effectively become administrators and distributors. Exploiting control over Umbar and ships would solidify their power. Each owner would be able to monopolize the transportation of the crops to market. Which means they could force the sharecropping former slaves into selling cheap. They would then be able to sell at market prices, which would grant high profits. This then would allow them to control access to credit. Thus debt becomes another tool for control.
The Southron allies would need to be protected, as it is unwise to alienate those forces required to stay in power. As long as they keep their privileges and are not treated too harshly, this should not be hard. After all, a successful conquest of their homeland they engineered would leave them little power base – thus they would be too timid and weak to act independently. There is likely to be assassination attempts, raids and other guerrilla warfare tactics used by the population against Gondor and their collaborators. Conquest would be easy, but the occupation much harder.
For the Harad, things would have to be done a little differently. The Prince talks about occupation quite a bit at the beginning. One of the things it talks about is populations that are similar to the people taking over are less likely to rebel. Also less likely to rebel are those used to a monarchy as their government. While I have little information on the two societies, what I do have suggests a more centralized government and urbanization in Rhun compared to Harad. This means that the Southrons are not merely being forced to change rulers but also their culture. At least to a larger degree than the Easterlings.
Next, there is the problem of coalition dynamics. Again, going back to the Prince here, those who become collaborators are likely motivated by their desire to improve their position. The problem is the costs of war and occupation leaves them worse off. Not only will foreign troops be moved into the area, but taxes are likely to be increased. So, some of the allies used to get control are going to switch sides at some point. This might not be too big of a threat, but it does make occupation much harder. So the colonization I propose for Rhun would not work as well.
What replacement do we have to colonization like in the Rhun? I think that this area would be a perfect place to copy the Spanish imperial model. Elites in the Spanish owned South America held large estates that had mines, plantations and/or factories on them. Typically the elites lived in the more urban areas and left local managers in charge. Related and preceding the land grants that formed the haciendas were encomiendas. These were the right to the labor of a specific number of native people that the lord was responsible for converting, protecting, preventing from rebelling, and constructing infrastructure to support. Also used almost from the beginning was the use of reductions – or settlements modeled after Spanish towns created by forcing native people into them. So, yeah, grant large amounts of territory to nobles with the right to exploit the labor of the residents for profit so long as they improve the area and provide security.
Pelargir would become extremely important for trade. The river would provide essential transportation links for goods and people coming in and out of the Harad interior. Then, this same river would connect to Umbar. The only place that could then connect Umbar's increased commercial traffic to the rest of Middle Earth would be Pelargir. We can assume then that this would cause a ship building boom. Increases in trade based jobs would increase in both cities. As would the size and profits of the merchant class.
Yet, civil war may still arise for another reason. Even if we do not accept the idea that the Southrons and Easterlings are inherently violent and warlike, the colonial practices are likely to produce justifiable anger. Especially if policies are enacted to break the cultural connections to Sauron. Morally, this would be a good step, as it would reduce slavery and human sacrifice. But no culture would accept that. Add in the chaos of above and the outlying regions will rebel. This is what happened in the Russian Civil War as well. The collapse of the Ottoman Empire had similar elements with the Arab Revolt.