I managed to get a Zen and am testing it with my quadstick.
Running 2328 on the quadstick and the Zen's FW is updated to 2.0.11. Its output is set to PC/mobile (which is "xinput"). The Zen is connected to the PC with both its output and Prog ports. The quadstick is connected to the Zen via the USB-B port.
Emulation modes 1,3,4 & 7 all worked with the Zen. Emulation mode 4 is the "Boot in PS4 USB mode" setting. With that setting off, which is emulation mode 0, the Zen only sees the mouse interface and treats it as right stick. In mode 4 it sees the mouse interface but that doesn't cause trouble.
Turning on the Titan 2 PS4 flag is also a good idea to prevent the quadstick from trying to authenticate.
I tried both front Zen sockets (A2 & A3) with the same results.
Next, I connected the QS USB-B port to the PC, turned off USB-A host mode, and connected the QS USB-A port to the Zen A3 port with an A-to-A cable.
After powering up the Quadstick with "Boot in PS4 USB mode" turned on, then switching from USB-B to USB-A, the Zen would not respond to the QS controller outputs, which I think was your original complaint. (It would respond to the mouse output but we don't want to use that) Changing the emulation mode to 1,3,4 or 7 would make it work.
Here's what is going on: When the quadstick powers up in USB mode 4 while connected to a PC, it detects that the host is a PC, sets an internal flag, and switches to mode 0. Once the PC flag is set internally, switching the connection from port B to A, the PC flag is not reset so the code thinks the Zen is still a PC and the Zen only sees the mouse interface. Changing the emulation mode to 1,3,7, or even 4 again, will let things work. When the emulation mode is changed, the PC flag is reset, so mode 4 will work but as long as you are setting the emulation, 7 would be my recommendation to get the fullest range of outputs.
Regarding the Zen not running scripts: Make sure the Zen mode is "Operational" and not "Tournament"