The word found in 2 Kings 11:4 is כָּרִי֙ Cari while the word for Karite is קראים
As you can see, these words have different roots, one being כר (ram/body-guard) and the other being קרא (to proclaim/the Scriptures):
כָּרִי adjective, of a people probably = Carian, only with article as substantive collective Carians, name given to foreign body-guard of king. - BDB
These people are also known as Cherethites, as in
2 Sam. 20:23 which also mentions them.
J. Strong thinks, "Perhaps an abridged plural of kar in the sense of leader (of the flock); a life-guardsman -- captains, Cherethites (from the margin)."
So, alas, no relation in regards to the word, but your question is nonetheless intriguing. As far as I know the Karaim 'sect' originated in 8th C. Babylon, but the Scripturalist mindset has always been around, and is one Yeshua taught when he quoted Isaiah 29:13 in Matthew 15:9. However, there was no sect called Karites at that time.