That's quite steep TBH; but if you haven't the tools/time/patience to do the swap then fair enough
That battery is here via Tayna:
https://www.tayna.co.uk/car-batteries/bosch/s5a13/
£170
A Creator C410 code reader via Amazon is c£55 and worth having in your arsenal IMO; it'll also register a like for like battery
https://www.amazon.co.uk/Creator-Bat...s%2C165&sr=8-4
Both still a good saving over what the AA have charged
Therein though; still hell of a lot cheaper than what BMW would charge!