In conclusion, I'm starting to think I chose the wrong character
for the game. Our 'rebel' party is mostly neutral
This shouldn't be a problem. Urist just needs to toss out enough plot threads that everyone gets lost in the layers. So far everything's been a bit simple.
Hypothetical scenario:
Pick a neutral player in the party. We'll call him X.
1) Rebel player gets a message from base stating that X is suspected of being an Imperial agent, and requesting said Rebel player monitor and attempt to corroborate it with definite proof.
2) Imperial player gets a message from base that X is a Rebel spy, and requesting said Imperial player to try to win X's trust in order to infiltrate his organization.
3) Have player X be privately contacted with a totally unaffiliated job offer that happens to require the party visit a contested planet.
*Poof* now everyone has a motivation that is consistent with their character background: The neutrals are happy to take an unaffiliated job because it means money without risk of alienating anyone. The Rebels are happy because they feel like they're doing good deeds. The Imperials are happy because they feel like they're working for the Empire. The party also benefits from the situation discouraging players from being actively antagonistic. The Rebels aren't sure that X is an Imperial. The informant was clear that he might not be, so they'll be busy trying to get "definite proof" which is unlikely to happen because X isn't an Imperial. Meanwhile, the actual Imperial(s) in the party get to feel like they're chasing Rebels, but they're under strict orders to
not turn in the player X who they believe is a Rebel, because they're supposed to be rooting out the broader organization rather than the small fish.
Then we simply apply more layers.
For example, the party arrives on the contested planet, and Rebels on the planet openly contact the party for a simple passenger transport deal. Nothing very scary, but they make it clear that they're Rebels. Rebel players are thrilled because now they're actively and openly working for the Alliance...but their characters don't say so because they've been told there are Imperials in the party and they're trying to gain their confidence. Imperials in the party are also thrilled because they're now worming their way into the Alliance for later betrayal, but they don't turn them in right away because even though they're Alliance passengers, they're obviously not important enough for the Rebels to try very hard to be secret about it. And the neutrals are happy because they're getting paid.
Add more layers.
However, when the ships arrives at its destination, it comes out of warp within sight of a Star Destroyer which immediately tractors and boards, because the "Rebel" passengers are actually Imperial spies, who try to turn in the party for being Rebel agents. The party members are put into separate cells, where each is told something
totally different by their interrogaters in the form of private whispers from the GM. For example, a neutral player can be told by his interrogator that they know he's a Rebel, but that if he helps to infiltrate his Rebel base (which he doesn't have because he's not a Rebel) that they'll pardon him. A Rebel player might be told by his interrogator that his background checks out, and that an Imperial player in the party is a suspected Rebel spy, but after a full inspection and interrogation, they don't have enough information to prove it so they're going to let everyone go. An Imperial player is immediately identified as an Imperial, is asked for a debreifing, but it turns out that the orders described in 2) above cannot be corroborated with base, and so they're now under suspicion. But...since they're still an agent, they'll let the party go and the Imperial thinks that it's solely because of them that the party was released.
Everybody gets a different story, none of which agree with each other, and then the Imperials drop everyone back on the ship and leave. Now, everybody has a massive pile of conflicting information and agendas, each of which they have reason to keep secret, and many of which are based on faulty information.
So with having all been set up, rather than providing resolutions, we proceed to immediately
add yet more layers.
Right now everything is too straightforward. GM just needs to apply plot threads until nobody is really sure which side anyone else is on or what anybody's motivations are. Sooner or later players will start trying to figure out what's going on, which will be difficult because there
isn't anything
really going on, but in watching us talk the seven of us will probably come up with some entertaining theories as to what's going on, and the GM can incorporate the ideas he likes.
This is, of course, just one possible solution to the problem. There are others. And since I've described this in detail it might be better if the campaign did something else so as to not be predictable. But the point is that faction affiliation really shouldn't be a limiting factor here. The game can easily be made to accomodate it.