Sorry to disagree, but a neutral is essential for a GFCI. A GFCI monitors the currents in the hot and neutral wires.
It monitors incoming and outgoing current... This can be done without a neutral in a straight 240VAC circuit, as it is done on heating elements, hot tubs etc...
How many 240VAC heating elements have YOU seen with a neutral wire? Hot tubs?
Maybe this will help...
There are two types of two pole GFI breakers, straight 240 volt, with no neutral lug, and 120/240 volt with a neutral lug. not all manufacturers make both, so this is something to consider before installing a panelboard. The 120/240 volt works for both applications.
A 240 volt load, such as a heater element, needs no neutral, but must have a n equipment ground. A straight 240 volt works simply. It will monitor the current flow in both hot legs. If the current flow differs between both by 5 milliamps or more, the difference is assumed to be leaking to ground, and the CB will trip.
A unit that needs 120 and 240 volt needs a neutral. For example, the heater needs 240 volts, no neutral, and the current in both hot legs will be the same, let's say 10 amps. If only the heater is running, there is no current in the neutral, only 10 amps on each hot leg.
Now add a 120 volt pump that draws 5 amps. It only connects to one hot leg, so it impose 5 amps on one hot leg and you can measure 5 amps on the neutral.
The one hot leg the pump is connect to sees the 10 amps of the heater plus the 5 amps for the pump, while the remaining hot leg stills only sees the 10 amps of the heater.
I believe it is this type of scenario that is concerning you.
They contain intricate and intuitive electronic measuring circuits that must monitor both hot legs and the neutral, and do the all the comparisons, by knowing that one hot leg, while has more load on one due to the pump, adds it, compares to the neutral, and as long as it sees the proper current , no tripping occurs.
Once the current flow differs 5 ma or more in any hot leg, and does not see the current flow in the return, whether the other hot leg or the neutral, it then assumes a leak to ground and trips.
This is why these breakers are so costly, in the range of $130.00 each.
Contacting the manufacturer is a great idea. I know, as I have done this many times before for various reasons, due to proprietary information and industrial secrets, you probably will not get a much better answer.
Plus, on your initial call ,I guarantee you will get some layperson in a call center reading a generic script for answers, and you will hang up not knowing any more than your when you first called.
Be patient and persistent to speak with a product engineer. He/she will do their best to explain the operation without giving away trade secrets.
I find that GE and Square D are fairly useless getting tech info. Siemens and Eaton ( Westinghouse and Cutler Hammer) are much better.
Sooooo, in the OP scenario where he is running 3 wires to the kettle heating element (2 hots and 1 ground) he does not need a neutral for the GFCI to work, again... scary thread we have going here.