I'm new to this forum, but have been playing with speaker design for a long time. One thing may affect the 'hearing the tweeters' in the sound is the front of the cabinets. It is not often given serious consideration, but some sound from the drivers propagates across the surface of the front of the cabinet. When these surface waves hit the edges of the cabinet, they encounter an 'acoustic impedance mismatch', which causes a portion of the surface waves to reflect back, and a portion to be radiated out into the room with various delays.
Rather than arguing about the audibility of this effect, it is easy to test it by dampening it out. An easy way to do that is to use a towel. The fuzzy terry cloth tends to dampen surface waves. Just take a towel, cut holes in it for the drivers (make the holes maybe 1/4' bigger than the radiating surface of the drivers), and stick it on the surface of the cabinet. For better results, make two layers.
For such a big cabinet with multiple drivers as the CBT36, try making the terry cloth with holes in it for the drivers cover just a foot or two of the cabinet, then listen up close to the damped portion and undamped portion of the cabinet. Try pink noise if you can. You'll probably hear a difference, and may like the sound. If so, you can try doing this for the whole cabinet. The effect may be dramatic enough that you may want to tweak the electrical EQ. But the net result should be that it sounds less 'speakerish'.