Battery charged ok, so not a failed battery (I have a CTEK charger that checks battery health). Probably flat from sitting at the dealers for a few weeks followed by short journeys)
I don’t have the radio code card, so changing battery may be tricky, though with jump leads it should be possible to change the battery without interrupting the circuit...
Working on a battery in a tight place like a Jazz is tricky enough, but while trying to keep the large jump lead clips on the battery clamp it can be very difficult ( although you only need positive on battery clamp , can you can clip negative to earth on engine or body ) - the large clip gets in the way when trying to undo battery clamp ( and the positive clamp has a insulating cover over it ) - if the positive jump lead or battery clamp it is attached to touches anything metal under the bonnet there will be an almighty flash.... and if the jump lead clip falls off you have lost power to car electrics anyway, but as above if the jump lead clamp falls off and touches metal - bang...
At the moment the age of battery on your car is an unknown, and anyway it has possibly sat on a dealers forecourt in a low state of charge for a long time then it is damaged anyway ( sulphated ), no matter what the C-TEK says. Get a Yuasa silver from Tayna ( for under £60 delivered in a couple of days it is a steal ) and you will have peace of mind for a least 5 years and probably longer. I think punctures have replaced batteries as the roadside emergency service biggest call-out nowadays, but failed batteries are still high up on their list.