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Maybe I should explain as my previous comment was a tad cryptic.
The way the fob system works is that the fob unit transmits a coded frequency to the remote control door lock receiver which is mounted in the rear or the car. Once received it sends a signal via the serial data circuit to the BCM which unlocks the doors. The receiver is shared with the tire pressure warning system. The system is calibrated by tuning the fobs to the receiver. The 97-00 cars operate at a different frequency to the 01s and onwards. The export models have a 3rd frequency.
The C6 fob would need to operate on the same frequency and be able to calibrate to the C5. In your case thats the newer version for the 01. I don't have any details of the C6 fob but I'm guessing the chances are very slim.
One member produced a foldaway blade type fob a while ago but there were problems with delivery and his vendor status was canceled. I think you can still get them through some vendors.
I don't know why someone doesn't just make a new casing for the fobs. It's not like it's that big of a deal. Obviously it's expensive to tool for the first set, but IMO it'd probably be a good seller.