Just repeating what I've been told on courses- alternator's generate a surface charge on modern batteries (which the systems run-off), which disappates into the cells once the engine stops. This is why you could jump-start a knackered battery, drive around all day and it'll start ok, but as soon as it's left for a length of time (overnight) it'll be dead again.
Take your issue with your sister's Panda- on trickle charge it did nothing, all the charge soaked away. But give it a full deep-cycle/pulse charge and it was able to maintain it's volts/amps.
Once fitted, the IBS monitors the battery condition (voltage capacity, drop, etc) over the next 5 starts with a gap of at least 8 hours between them to determine the battery's ability to perform the repeated stop/start operations. If it's starting from a weak first-charge, you'll have system problems.
Usual training procedure is to send at least one Tech from each Dealer on the course, although some short-sighted Dealers merely see the cost involved and won't let their Tech go.
The course notes gave a detailed list of what various warning light/message combinations indicate, though I don't have them to hand, I think you're right in saying a Battery related issue would normally bring-up 'S/S Unavailable'.
Unfortunately as with all new Technologies, it takes a little time for solutions to be found for common issues that crop-up, and the symptoms you have suggest a software glitch which is out of the Tech's hands as it relies on Fiat/Bosch identifying the cause and finding a solution (whether it's a simple reprogramme, or requires modified parts).
At what part of the 'Start engine, press clutch, select Reverse, raise clutch, drive backwards" sequence do the warning messages occur?