You want an idea on how to do this...
Go with a mixed up shiplist. Each race has their usual stuff (or a specialty-ship heavy subset of it), and a mixture of "enemy" ships (line / command) added in.
The trick is that "native" ships have to stay
in hexes of their own color, while the "enemy" ships really need to be out in enemy (or neutral) territory.
Now, an example of all this:
The Klingons. Their shiplist would have the usual Fx, Dx, Cx series, possibly heavier on -D, -V -G and -C/L variants. Meanwhile, there would be a smattering of Fed and Hydran line (command too?) ships in the Klingon yards.
A pilot in a Klingon hull would have to stay in red Klingon hexes, while the Klingon-owned Feds / Hydrans have to spend as much time as possible in the neutral zones / attacking enemy space. They should only return to "home" hexes for a resupply run / sale, and if there is a 90% available "on demand" repair / resupply mission that can be run often in enemy space, the only time a privateer should "go home" is to sell the ship.
As far as RP goes, here's the rationale:
As you said, war really ain't an option. Therefore, if a native ship flies off into a neutral zone or "enemy" territory, there's that act of war that everyone's trying to avoid. So, native ships need to stay in hexes of their own color.
Meanwhile, the privateer fleets are off in enemy space. Since they're flying "enemy" designs in "enemy" space, they'll look (at least to the natives of a hex) that the current government is a corrupt morass with hideous allies, and they would eventually seek the protection of the other team, who would fly their native ships in to reinforce the now-appropriate colored hex (explains the hex flip nicely, doesn't it... :p).