First of all, the Karen don't speak Burmese, their language is Karen (which comes in a variety of fairly dissimilar dialects; by far the most prominent of which is Sgaw Karen, and the second most common being Pwo Karen). Most of them also speak Burmese, but it is their second (or in some cases third or fourth) language.
Secondly, I understand the bothersome connotation of KK especially for those of us with a memory of the olden days in the Civil Rights Movement, but I know of no way to soften it except to spend more time thinking of the current KarenKonnection than you do of the old Klan. Eventually the good will outweigh the bad. I too wish the Binkleys had used C for Connection, not so much because of the Klan connotation as because it seems to me the Konnection bit is not ideally conducive to encouraging correct ESL acquisition by the Karen. But that's a minor point.
BTW the Karen language uses a version of the Burmese alphabet (technically an , not an alphabet), so there are no "K" letters in it. The Burmese alphabet is called the "apple script" because so many of the letters look like apples with various bites taken out of them. Here is a chart of some of the main consonants in this script. The one highlighted in blue background is "k", but looks more like a rounded Celtic M.