I'm with the folks that say the reflector isn't doing anything except catching spill from the LEDs and reflecting it forward, probably in the form of rings. You could try modifying the shape of the epoxy domes of the LEDs (kind of like the LED "bulb" in the old Dorcy Cool Blue)-- it would be fun to test different shapes: flat, concave, V-shaped (you could drill into the dome to try to make the light go sideways into the reflector). Or you could try painting the dome with silver paint.
As a confirmed Big Lotser, I sympathize. Check out their little 3-LED AAA flashlight for $3.50 while you're over there looking for materials to modify things. Not superbright but clever and well made.
Walt "Incredible Cheapskate" Brand