The version 2 triaxis and beyond is capable of CC changes, along with the Red 1 board replaced with the Rec or fat.
I have run a triaxis for 15 years, but I never got into the CC of the Triaxis. The reality is that for distortion, you're going to most likely end up with a ld2 channel preset and then have a clean preset. The ld2 modes clean up so well by rolling back the volume on your guitar, there is no need to do it via CC. If you want a lead guitar patch and a rhythm patch, the changes are instant by pressing the midi controller, you won't need CC there either.
As to assigning midi patches, the first 20 on the triaxis are factory reserved. Starting with patch 21, create a good dirt rhythm, personally, I like ld2 green (mark iv). From memory, 8g 7t 4m 2b 0 8dist 4.5master 4v, my lead sound is ld2 yellow, 8 6 4.5 4 0 8 3. A little thicker and closer to the John Petrucci settings. Note, the higher the treble setting, the more gain you get, but the lower the bass needs to be. It's not uncommon to see people running these with the bass at 0.
So programming the triaxis. Preset 21 and patch 21 are the same. Program your controller to that first number 21 (that is the actual midi number that the controller board cares about). So my first button on my controller is set to 21, my TC Gmaj ii preset 1, set to light chorus and a little stereo delay on this patch. My next button is set to the same triaxis patch, but my TC preset 2, with flanger to do VH type stuff. Button 3 is my lead tone, so Triaxis patch 22 set to preset 22, ld2 y. My TC is preset 3, more delays, etc.
As a footnote, I believe that I am using the TC as the first midi device in the chain with the through going to the in on the triaxis. It's been forever since I programmed it, but I believe you can make the TC midi settings look like preset 1 = 21, preset 2 = 22 or similar. That means when your foot controller button 1 as a default is midi program 1, the TC changes to preset 1, but sends a signal of 21, which the Triaxis needs. A hell o a lot easier than the Triaxis' system or the midi foot controller.
Basically, I have five sounds built into the triaxis, and then set the first LCD on the triaxis to what ever corresponding midi patch I need. So that basic dirty rhythm might be set to 21 21(the actual patch), 22 21, 27 21 and my lead might be 23 22(the actual patch) 24 22. Again, the first number is what is being received from the TC GMaj midi.