The idea is lacking in game design insight and conceptual soundness. There's no strong drive for new mechanics, content, or otherwise that I can see in your proposal. If stapling the Pirate Factions to the existing militia is your proposal, why not just join an existing militia and not tie down developer resources?
Because as a Serpentis Loyalist i would like to support Serpentis, not the Caldari State. Sure you can argue helping the Caldari State indirectly helps Serpentis by disrupting Federation efforts but so does regular piracy and strip mining belts in high sec.
The difference is that including the pirate factions in FW makes FW more interesting, allows pirate factions to profile themselves as more then belt fodder and gives us pirate faction supporters a way to get those lovely loyalty points by shooting players instead of npcs. (fffuccckk shooting npcs)
The less sound method would be to develop a unique set of conditions for each of the Pirate Factions to influence lowsec instead. The Blooders are driven by Sani Sabik purpose and blooding motives; they can be rather pangalactic in their over all goals. The Sansha merely want revenge against the Empires, and although free-range Capsuleers are anathema to them, having them direct their carnage that way is useful.
Serpentis and Angels are ill-defined in ways that are conductive to galactic wide campaigns. They are the underworld of Empire civilization; they thrive in the illegal and criminal. The Serpentis particularly so; the Angels may have motives beyond that we do not know about yet. In that respect, though, they can be mainly based around stopping police activity. The Guristas are anarchists in general, so the Empires are a favored target, but their leader has a particular hatred of Capsuleers.
Obviously a system designed specifically for pirate factions would be preferred, the problem however kicks in with the fact that existing pirate faction content is hard to reach and thus not used a lot. Since not many players use it CCP is not going to dedicate any development time to it. Just look at all the various bugs they've ignored for years.
By including pirate factions in FW in a limited fashion, for example by not allowing them to capture systems but by only setting the pirate militias up with wardecs against the empire militias you expand on pirate faction related content with little to no effort on the part of the developers and you make pirate factions more accessible to the general eve population, which means it might eventually reach a point where CCP would be willing to spend development time on it.
As for motivation, most of the pirate factions have simple reasons to get involved;
Serpentis for example wastes billions a day on patrolling every belt in and around Federation space. Possibly trillions if you take into account the loss of crew life (and thus training) not to mention the complexes and bases they have to rebuild every single day. Apart from massive grudge Serpentis holds against the Federation they need to think about business; this business if hampered by both Federation forces and Capsuleers in general. If handing out some loyalty points to get more capsuleers blowing each other up and in the processes creating more chaos along the Federation border then thats pretty much win-win for them.
Sanshas invasion of highsec is a giant failure, all thanks to capsuleers. Its safe to assume they'd be quite happy enlisting capsuleers to shoot other capsuleers, the more time they spent shooting each other the less they spent shooting Sansha forces.
All of the pirate factions have one or more reasons for wanting to destabilize low security space.
Giving pirate factions letters of marque for killing the Militia of the opposite faction would be doable?
Judging by how the FW system is setup this shouldnt be too much work. The only coding required would be a check to prevent pirate militias from capping buttons and a way to select which militia screen you want to look at when you open the militia window.
The problem with letters of Marque kind of approach would be that there would have to be overt co-operation by the empires with the pirate factions and CONCORD acceptance of the pirate factions themselves as something that should be dealt or bargained with, which would kind of go against the idea of law enforcement in general.
It would be for the same reason CONCORD allows pirate factions to be factions and protects their sovereignty from being conquered. Since no IC reasons are ever given as to why CONCORD supports the pirate factions its most likely because of bribery, CONCORD has always been portrait as pretty corrupt anyway.
At the end of the day FW offers an opportunity to add content to pirate faction side of eve with very little time required by CCP. Content which will otherwise not come because its simply not worth spending an entire patch coming up with new content for a part of the game that not many people use and the majority of eve will be locked out of due to standings.
If adding pirates to FW draws more players into that side of eve then perhaps it will become big enough for CCP to eventually iterate it into a more unique system.