The small battery is charged by a "dc-dc convertor"/"charge pump" type charger which will be able to take power from the main battery ( and/or alternator if it happens to be running ), Jaguar call it the " Dual Battery Module". So best ignore the voltage of the small battery in your diagnosis.I'm currently struggling with self-diagnosis of the same "Battery not charging" dash message. At rest, the main battery reads whatever is left in it (11.9v in this instance), and the small Stop/Start battery reads 13.1v or thereabouts.
With the engine running, the Stop/Start battery reads 14.3v but the main battery's reading doesn't change. If the alternator is defective, what causes the voltage increase in the Stop/Start battery when the engine is running?
Perhaps coincidentally, but I'm not so sure, from the start of the journey until around when the dash went red with the message (10-15 miles later), there had been a faint whining noise much like a worn bearing, matching engine speed in pitch variation and most noticeable at or near idle revolutions. From then, any trace of that noise has been entirely absent.
It's worth checking that the battery terminals for the large battery are securely connected, that the negative cable from the battery is still secure at the bodywork end, and if you've just changed the battery that the thin wire that runs from the positve terminal on the large battery to the BMS module on the negative terminal is ok ( it seems it's a bit fragile and sometimes gets broken/damaged when the battery is changed. )