Attack on Lake Town

The Conflict

Prelude

On the way to Dale, the Easterlings would probably deploy one legion and Auxilia brigade to attack Lake-town. This would break the links between their enemies. Troop movements for the Free People would be hindered while the logistics of the Easterlings would be secured. Plus there would be the need to break the underground railroad like network that runs through the area.

Battlefield

This operation would run through a huge amount of territory in the north. Many smaller villages would be in the path of the legion as they move to take Lake-town. Each with their own militia defenses and Motte and Baily fortifications. None of this would be hard to overcome. But it would still have to be overcome.

The Engagement

Once the Easterling legion begins its advance, it will start to run into local resistance. The main issue here is the many small villages they would find along the way. The legion would need to eliminate the threats otherwise their supply lines would be exposed. So each village would be surrounded. Then burned. Typical assault tactics would not be used as occupation is not the goal. The population is if no use - if not an outright threat. The locations are also of no use themselves. Perhaps Lake-town would be useful for them to retain long term, but not the random villages on the way.
So their advance would be a series of massacres of civilians and looting/burning of villages. The response would be a collection of diffuse militia units engaging in a traditional hit and run resistance movement. Each village would take a day to take. The militias would be able to exploit this to hit supply lines and bait the enemy into sending out parties to find the militia units. Others would carry out reprisal attacks and sent to escort supplies. The result is the sort of dispersal that works to the advantage of the militia. The Easterlings would not be defeated for a while. But it would be a death by a thousand cuts.
By the time they get to Lake-town, they would have been weakened over the course of the march due to low level losses and dispersal. But they would be reinforced by the time they get there by the shattered survivors of Dol Guldor. I suspect that the total forces available after these reinforcements and losses from both groups are factored in, they would have 3 legions and 2 full auxilia brigades.
This is where the power of Lake-town's location demonstrates its power. They would need to surround the whole lake in order to fully besiege it. Otherwise, the inhabitants could just use their boats to run supplies in from other shores. Direct assaults would be impossible without a lot of prep work. The Easterlings would not have enough of the right boats to pull that off. So they would attempt to build their field works around the lake while figuring out a game plan for the main attack.
One annoying issue. Its not like the rest of the Free People just disappeared at this point. The attempts to build the field works would meet escalating hit and run attacks as the militias still operating move into the area. They would also probably continue to get civilians out and soldiers and supplies in. Then there would be the eagles, Mirkwood Elves, Moria Dwarves and the small force of Rohirrim.
The result would not be nearly as one sided as at Dol Guldor. First, the army of the Free People would be much smaller. Second, I suspect that the prep work would be a bit more in place. The fact they are coming would be less of a surprise either. But the loss of morale from the previous battle at Dol Guldor and the Northmen guerrilla war would take its toll. Instead of a burning forest trapping the Easterlings, the lake would hamper escape.
But this is where the value of the professional military force that is the Easterlings would show its full strength. Part of the value of Lake-town is the ability to use the river to supply the assault of Dale and the Lonely Mountain. There is no way in my mind that they would not have the organization needed to load these troops into ship transports. Essentially, river transit is already established as the primary means of transportation in medieval societies. Then, the Easterlings would already have logistics operations based on rivers. The full importance of that would become clear in Cair Andros. But here, it seems logical to assume that the Easterlings would be set up to move their ships into the area.
Clearly, these would not be near as good as those operated by the Corsairs. But one does not need them to be in order to move troops up a river. The eagles would continue to be an issue. But most of the opposition would be land based forces that would not be much of a threat - relatively speaking at least. So what I think would happen is the Free People would show up and have issues getting past the prepared defenses of the Easterlings. But they would be too powerful to not eventually win. Especially with the eagles. So the ships meant to help with the logistics of attacking Dale would serve as emergency evacuation transports.
These guys would probably first be used to burn Lake-town. Patience and time would be low. Nor could the risk of an enemy base near supply lines be tolerated. Meanwhile, the Easterling soldiers would maintain their defensive positions until the ships are available. Shorter and shorter lines would probably be constructed as one gets closer to the lake to allow for fall back positions. As more troops get evacuated, the remaining ones would retreat to these shorter lines. This would allow them to retain defensive power for the duration of the evacuation.

Outcome

Essentially, the whole area would be torched. Many northmen villages would be lost. But the Easterling armies would be hammered too. And for little gain. The Northmen were hardly the greatest military power and they would be the only power knocked out thus far. Mirkwood remains intact. Troops would be lost and some of its territory burned, but this would most likely just increase resolve.
To be fair, this campaign was not meant to be a decisive battle either. The key Easterling objectives were met. They eliminated a long running threat to their social order through shutting down the network helping escaped slaves. Their ability to move men and supplies up were enhanced by the clearing of Lake-town. All in all, mission accomplished. Just too many troops died in the process.