Ugghh, so much conflicting info, especially it seems from Fiat employees who clearly are not really sure themselves.
What I can take away from all this is that the battery within the pebble is a standard CR2032 (as evidenced by blindoracle who has seen it with their own eyes) and so is certainly not rechargeable. Also, as my car is still in warranty, there is a chance I can get a replacement from Fiat (as evidenced by IanML who had one replaced). The way forward is clear: Try Fiat for a replacement and if I have no joy there, rip into it myself as I have absolutely nothing to lose.
Thanks all.