Convertibles weigh about 250-275lbs more than their coupe counterparts, so they typically have slightly stiffer front springs. Not sure about dampers and swaybars. I've driven both the ZL1 vert and a 2SS vert and there is a significant difference in handling, but I couldn't tell you that it isn't just the stiffer springs, tires & MRC damper tuning.
If your car has an FE2 or FE3 suspension, then you can probably bolt on a FRONT FE4 swaybar+bushings+links and take advantage of the 3 position adjustability on the FE4 bar. If your car already has an FE4 suspension then try moving the end links to another hole to stiffen it up.
I don't know whether the A1Z package on the ZL1 1LE has a different or the same swaybar as the standard FE4. A quick search did not turn up a lot of different stabilizer bar part #s... so there may not be a lot of choice from the OEM bin for the FRONT swaybar. There are more choices for the rear of the car, but what will fit is limited by your differential.
More info here:
https://www.camaro6.com/forums/showthread.php?t=542333
I'm sure the folks at
BMR, etc. will have a solution for you if GM doesn't.