Just an update - I have been running a GT/22 and Axe-Fx III together with Mr.E's settings (on the GT) for 5 days straight now. I'm also running Axe-Edit at the same time. When I walk by it, I change presets a few times and change scenes a few times using the GT. So far, I haven't seen any issues with incorrect preset numbers.
I looked at the previous posts again, and I noticed the number 127. Is it possible that it's actually 128? That's the size of a MIDI bank. If CC#32 is sent to the GT, or CC#0 is sent to the Axe-Fx, that would shift the bank number. For example, if you send CC#0, value 1 to the Axe-Fx, then all PC messages sent after that will access presets 128-255 instead of 0-127. The only way to get it back to presets 0-127 is to send CC#0, value 0.
The GT does this automatically to allow access to all 768 presets, but maybe something else sent a CC to shift the bank? Is that possible in your setup?