Linda Lee feels... fake in a way most secret identities usually don't. It's a name a teenager that just lost everything picked randomly to fit in with the strange people her cousin was leaving her with.

Linda Danvers is much better, because that's the name said teenager uses after she has found a new family with two people who love her. Then again, having her go by that name these days would mean making the return of the post-Crisis Linda Danvers more complicated than it already is (I'm convinced they'll bring her back in some way eventually).

So yeah, having her go by Kara in her secret identity is perhaps too suspicious, but her costume is far more believable than Clark's anyway, so I don't see it as something problematic.