AFAIK, when you plug in the power on the Z, it supplies 5v to the ports instead of the usual 3.7v. Maybe your card can't handle the stronger current? I don't have any documentation to point you to on this one other than the voltage listing on the back of the unit, but here's how I tested this:
I have a Kingston 2gb pcmcia hard drive. I tried it using my compact flash to pcmcia adapter and it seemed to work, but when I started playing a song from it, it would stop and power down. I figured the power on the port wasn't enough to keep it going on a sustained level. So, just to test my theory, I plugged the good old ac in and was able to use the drive indefinitely. Based on this evidence, I have to conclude that it does in fact supply 5v to the cf slot when on ac.
Now, knowing that, I would say that your card is only a 3.3v card and the 5v overloads it, causing it to stop working. I don't know what to suggest to you other than getting a cf to pcmcia adapter and using a compatible laptop card that supports both, or looking for a cf card that will. I have an OEM Proxim pcmcia card I was using with OZ, but it doesn't show up in the Sharp ROM(see the networking section if you have any suggestions) and it worked fine one either battery or ac.
Peace,
Todd