Playing the beta (and needless to say really liking it) I've come up with a couple of things I think would make capital ships even neater. Or one fairly big thing, but with subpoints:

Event-based leveling! My thought here is that in addition to the normal xp-based advancement, it would be really neat if capital ships could gain abilities based on what they've done in the game, like a certain amount of damage done to planets (or planets razed) would give a bonus to future planetary bombardment, along with maybe a little death's-head decal on the ship that won it, and possibly some sort of badge on the info screen ("Destroyer of Worlds" or something similar). You could have similar badges for cap-ship destruction, frigate destruction, fighter destruction, maybe breaking tac structures and logistical structures into two categories, all with different decals and subtitles. And maybe different grades, too (now that I think about it, I'm totally stealing this mechanic from Total War).

Other triggering experiences could be things like taking out a/the pirate base (depending on how that gets implemented) or being the first to visit a certain number of artifact worlds (although keeping track of that seems potentially problematic).

I'm not really sure how you'd assign credit, though. Like maybe 50% of the "points" for anything that happens in a gravity well would go to the flagship or else the highest-level ship there, and everything else would be divided between the remaining caps.

Also, I think artifacts for cap ships could be really cool. I imagine them being implemented as a unique ability granted to the first cap ship to visit an artifact planet.

I see all this ultimately combining in a given game so you have, say, your flagship "The Mighty Brick, Destroyer of Worlds, Classic Fleet Champion" with skulls, mushroom cloud stencils for each destroyed planet, and a bunch of enemy ship silhouettes visible on extreme zoom, a real bruiser against enemy planets and cap ships, and for that reason that much more important to keep relatively safe. Which basically is like cap ships as they are now, only moreso.