I've decided to just assume the fighters are pretty much to scale in the game, and that the in-fiction explanation is that they simply can't be made smaller and still carry all the systems, weapons, armor, engines, and reactors to power it all that they still need to be useful. I.E. they could be made smaller, but then they couldn't carry weapons big enough to scratch anything, or have enough armor to not just disappear if they get to near any debris/weapons fire/space dust, so fighters being the size of a house ends up being the most efficient way to make 'em.
I know, it's silly, but it's also novel, and I'm going to keep believing it until we get any official word otherwise.
As far as kilometers long goes, I doubt the current capital ships of Starfarer would come close. Maybe a few hundred meters for the big ones? I'd be surprised if any of them broke 500 meters.