I think it should be pretty safe to go through the channel with a kayak - certainly much safer than to go around Denmark with a kayak :-)

The rule is that one should go by engine through the channel, I suppose that excludes kayaks. There is also rules that one is not supposed to stop anywhere, or take boats in and out of the water along the channel. I believe if I went through the channel in a Kayak, I would have stopped and enjoyed a break together with some of the biking tourists :-)