ZPL does support GS1, but I'm checking if Zebra Designer does. Are you using DataMatrix or QR code?
Can you export the label to a file? Set up the driver to point to a file rather than the printer port in "Printer Properties" -> Ports. Then you can see what the designer is exporting. If you post it or the .lbl file we can take a look.
ZPL does not support concatenating variable data into single bar codes. The GS1 wizard in ZDesigner does the concatenation in the software, which is why it's static output. You will have to set up a standard DataMatrix or QR code in your label format with variable data. Then you will concatenate the GS1 data in your software.