The supercardioid pattern isn’t the culprit. Schoeps calls its cmc641 a supercardioid. The issue is that the capsule of the the shotgun mic is at the base of a long slotted tube. So the capsule is further away than you think and the slotted tube while making high frequencies very directional, still allows mids and lows to get into the mic. Here, watch and listen to with good headphones or monitors. this: http://gallery.me.com/tyreeford#100038
I’d use the lav, but try the shotgun if you can set up in a space with few hard flat surfaces and if you can get closer than a foot to the subject. Split track it with the lav on one channel and the shotgun on the other.