In 2008, he was the head of the Republican Senatorial Campaign and his work was at best incompetent.
He did it again in 2010 and he completely sabotaged the Republican effort. The GOP would have had at least 50 seats in the Senate without Cornyn working behind the scenes to prevent it.
How do you explain how he funneled the most money to senate candidates that became Democrats. And Cornyn was supporting them even after they switched.
He still bears the name “Republican”, so RINO still applies as a start at least. Other epithets can follow from that, such as traitor to conservatives as you point out.