Many thanks all, just in our last top up before the final stint home.
The car is leased and fully maintained so will be heading to the dealers for a new battery (or an explanation as to why not...) in the next few days.
I'm 100% sold on the booster pack idea, at this stage is rather try that than wait hours for the AA. Just need to hunt a suitable unit.
Regarding using the app, in all honesty, we tried it the first day once, after that it wasn't touched till early this morning, to try and heat the car up, at this point noticing that the app wasn't connecting, so it certainly wasn't battered.