I've noticed that while Kate torments Lizzie and Miranda, she normally leaves Gordo alone. It is normally Gordo who starts the Hostility with Kate.

It was Kate who politely asked Gordo about the city making software in Misadventures in Babysitting. She could have asked someone else.

Kate didn't seem to have a problem teaming up with Gordo in And the winner is. If I recall, she chose him.

Is Kate secretly jealous of Lizzie and Miranda because they get to hang out with Gordo?

maybe

I dunno... the evidence points in the other direction, too. She calls him Gor-dork in a few episodes. (That's mean, because personally, he's sexy.)