I don't have KM anymore, so I can't address that workout specifically, but when I was first starting out doing kickboxing, I had a workout that had similar drills in. It did bother my back some. I think part of it is maybe overstretching? Try doing it while keeping the kicks lower and see if that helps. I suspect that maybe the leg-held-forward moves might be more problematic (that position seems to put more stress on the lower back in me). Also, make sure to "tighten your corset" (an image that helps me keep my core strong) so that your ab muscles help support your back).
Did you do that portion after the rest of the workout? It's also possible that you strained the lower back a bit during the previous moves, and it didn't "kick in" (no pun intended), until that section.