IMO what is incongruent (nice word by the way) is that Corsair Eldar have a battleship not that Dark Eldar don't.
If anything what the DEs need is a light cruiser.
I find that, due to the speed and manoeuvrability of the DEs, they don’t have much trouble avoiding battleships and don’t need anything to go toe to toe with one.
Get in, hit hard run away.