This idea is separated into two distinct parts:

Part 1: Focused Strikes

Give EVE players (by which I mean corporations) the ability to spend FW LP to begin attacks upon a system.
The EVE side would choose a system to declare an attack on, and would be able to select a DUST Corp (including themselves) to be the group they "hire" for this mission.

The attack would begin in 1 hr, and would function much like a PC battle with the hired DUST corp counting as the team who is attacking. (In short, they get to bring in a squad or team, and organize that 5-10 min before.) At the 5 min mark, the rest of spots and the opposing team are gotten from the FW battle finder.

After a battle ends, if the contracted DUST team wins, another follow up battle begins, until 1) The system becomes Vulnerable or gets to 100%, or 2) They lose

This allows EVE corps to better direct the DUST FW battles to their own ends, and allows DUST corps that work with these EVE corps to get a healthy stream of battles where they can work with their own level of teamwork.

Extension: Add a payout pool (25% of the LP spent's Approximate ISK Value) that gets a rather large portion drained each victory to the winners. (or all remaining if it's the last battle in the attack)

Bare Bones: Allow EVE players to spend LP to cause DUST battles to happen in a system. (No selection of DUST corp)

Part 2: Planet-wide Campaigns

If Skirmish 1.0 ever gets readded (As in 1 MCC vs defenders) then FW could take on a bit different play style:

One faction owns a planet, and they can get attacked by the opposite faction who has X MCCs vs Defenders who has Y clones. (for example, 10 MCCs (And 800 clones) vs 1000 clones or something) at the end of each battle, Clone losses are tallied and subtracted from main score. MCC losses are similarly tallied. If Defender runs out of clones, then Attacker takes planet, like wise if Attacker loses all clones, he gets kicked off.

However, if Attacker loses all MCCs but not all clones, the battle goes to Ambush game mode, but with Attackers taking X% extra casualties due to Logistical Losses.

Battles would continue on the planet, and squads could have the option once a battle ends to continue in the same conflict. (Once the ability to join a fight from the end of battle screen is added)

This system could be in conjunction to the current set-up, and be a variant attack style on the FW front (AKA regular FW battles as is still can occur)

Extension: Battle-damaged MCCs. (Shields recharged to Full (of just by 50%), Armor repaired by 10% of max between each battle)

Extension 2: Also allow EVE players to pay LP or ISK to declare these type of attacks. (perhaps combine Part 1)

Bare Bones: Series of consecutive battles on one FW planet based on W/L of series, entire planet shifts all at once. (with the large gain of FW system control that comes with it)

Taking a system to vulnerable so quickly is absolutely not a good idea and would wreck the FW system in EVE.
I support the idea of being able to attack specific districts wholeheartedly, especially with more EVE involvement, but this suggested execution would end up being terrible.