I doubt that very much.
No. Replace the battery with a new, fully charged one and then get yourself a battery maintainer and keep it hooked up when not in use.
I did think initially, from your description, that it sounded like the alternator as it happened to me a few years ago. Good news is, it will be sorted at the dealership and you won’t have to worry about resetting anything as they should do it all for you.
You don’t need one Jon. Replace battery, reset park brake, windows and reset BMS and away you go. Or, hook up a temporary supply under the bonnet, change the battery and you shouldn’t lose anything.
An improvement sounds good though I would say you still have some air in the system. If you can live with it then fair enough, but do keep an eye on the temperature gauge if you go on any longer runs.
Park the car so it is pointing slightly uphill. Take the cap off the reservoir and then allow the car to run up to temp at idle. When the stat opens the coolant will flow around the system and the car will burp itself at the reservoir.
In that case the fusible link may have blown. If you look on the positive terminal you should find a single wire attached. This is the positive supply to the BMS module which is attached to the negative terminal.
The fusible wire is covered with a black sheath at the positive terminal and is about 3 inches long. A continuity test will prove whether it is intact or not.
If it has blown then replace it.
If everything is working correctly then you should hear one beep on double locking the car. You should then be able to place the fob at least 1 meter away from the car and not be able to open it.
If you have no warnings on the dash and the car is running correctly then putting it on a diagnostic machine is pointless. Simply maintain the car correctly and enjoy.