I like the functional names for Shivan wings. I hope that you use them, instead of making up something silly. Because, frankly, anything you do create will be a betrayal of our image of the Shivans. The less you know about them, the more frightening.
The only way I think you could effectively make up names for the Shivans is if you created a whole general ethos or culture for them. If you made the wing names in each mission spell out a stanza of poetry, or if you gave them goofy titles a la Iain Bank's Culture ships, that could work -- but only if you accompany the naming convention with an overall take on Shivan culture, i.e. they're warrior-poets or they approach their genocidal work with a light heart.
Otherwise, please stick to the functional names. And please don't use the hackneyed 'they like the name 'Shivan'' device. That's an awful, cliched SF tripe. How could an entire species not have a name for themselves and then decide they like a random one from an inferior race? You can use Shivan, but please, make it clear that it's a translation. Either that, or you could avoid using it at all.