Why not just write klementine?

Because c and l can only make one sound together. English has no word where cl says anything but k plus l, so the c is unambiguous the moment the l follows it, and the k spelling is never needed. It is the same reason clover, clock and clematis all keep a c that could be misread on its own and never is once the l arrives. Here the c also carries the name it came from: Clément, with a c, in French.