Well, I'm sure Firaxis could think of something clever -- such as a group hijacking a seeding ship or something. Though, it might be strange for them to be another generation ship since their knowledge of the seas and of that culture would be greatly diminished.
Also, going "pirate" may be a bit much for a playable faction. Especially when they need to construct many types of buildings within their colonies, conduct research, steadily grow their population, and all that other civ-like stuff. A bit of a stretch from simple pirate life.
Well, first to say I favour the priate faction! arr!. I think it will get more balanced for a North-South perspective, even at the cost of losing Japan.
As for the lore, why not go by actual (or at least as they told us) pirate-forming history? I would make pirate faction a discontent bunch inside de OSAS (Brasilia) that take one of their ships (they have several, vs the NSA that has only one) by mutiny.
This provides a war-trained leader, does not label any part of the world as being a pirate haven (altough it would imply it in some way). It explains how the seeding ship was achieved - and probably redirected to avoid Brazilia following them (to planetfall in the same world the own Rejinaldo is going, well... happens! - I also not see how Lena Ebner would choose the same world as Elodie: if not by mistake, it just means both Lena and this new faction do it consciously to piss off their nemesis

)
Back to pirates, they would be, of course, part of the marine branch in Brazilia's army and in the time from the mutiny to planetfall (what would mean they did the last part of the trip awake, they could modify their equipment to allow for cool things as sea landing). Their philosophy, similar from pirates will deviate to the orderly/disciplined "we are the leaders of the world" speech from Rejinaldo, to a more "let's live free and take the most of the new planet, arr!" - so they would improve their specialiation o exploration, artifact finding (extra explorer modules?) and pillaging (alien nest benefits increased) - this faction is willing to take risks in the new planet, instead of following a more cautious approach of a well-established faction. It could even have a bonus in some way similar to what Spain has in Civ 5, that becomes bigger as you focus on exploring the planet.
What if the last faction is an alien one?
I'd save that for a second (or third) expansion. (i.e. Beyond Earth: Early Contact). If done, I would like to have several alien types and factions introduced (i.e. 3 new alien races making for 6-9 factions), in a way you could a "balanced" scenario were i.e. for 8 players, you have 2 factions for each alien race - including terran) fighting for the planet. (You should then as well change contact victory, and probably add instead some diplomatic victory consisting instead on basing the grounds for a Galactic Federation. (Having 3 alien races would also be interesting to push each of them towards one of the affinity types - with Purity being
their Purity, not the human one, of course

).