More grind. What's not to like?

"Well, you're loyal, you've fought for our side in every match you've joined, and you usually top the scoreboard, but this time the rest of your team was a bunch of lazy slobs and you lost the match so we're only giving you a quarter of the usual loyalty points. Better luck next time."
Or you could gain loyalty points for being loyal and earn them based on how consistently you fight for one side to the exclusion of the other.
Completely replacing the ISK reward with LP is just insane. You might as well edit your post and just remove that completely.

Factional Warfare should reward LP.
Some things:

Losing battles for one side reduces the payout of LP you get from that side, up to a maximum of -75%

Winning battles for one side increases the payout of LP you get from that side, up to a maximum of +75%

If you win a battle for lets say, the Gallente, then fighting for the Caldari gives you reduced LP payout, and vice versa, up to -75%

You don't get ISK in a FW match, just LP(?)

So fighting for one side is more lucrative, but you have to actually win for it to be more lucrative.

Now, what will we do with that LP?
Well, I was thinking, what would make players REALLY want to play FW?

And then I was thinking: "Hmm, EVE has factional gear, why not Dust?"
Factional Gear = Better than prototype, worse than officer

By playing around, let's say, 10 matches?
You get enough LP to buy a set of Factional Gear. (Set being 10 items)

To buy that set, you also need ISK.
I was thinking around 2 million, that puts the price of that gear at 200,000 ISK a pop. (That's not a weapon or suit, that's just the gear like shield extenders or damage modifiers)
So assuming you get 200,000 ISK a match, that will require you to play 30 matches (20 for ISK, 10 for LP) to get 10 of a specific factional item.

Now, what about weapons and suits?
That will cost you double everything, to get 5 of those.
800k a pop, and 20 FW matches.

I would also limit each corporation to fighting for one faction. While we are merc no empire would hire on staff that are likely to go out and fight for the competition the next time a contract shows up.

Locking corporations into faction groups and LP stores actively encourages RP, as well as generating loyalty for that corps chosen factions....however im not say your shouldn't be able to defect to another faction just that LP should only be accrued, and LP stores should only be accessible one factions at a time.