I doubt the alternator is up the spout though i can test it I hope, volt meter across the terminals looking for 14V plus?
But I am always scared of doing something to cars with ECUs with a meter.
Simplest to take the batt to my local Motabitz for a proper test. they are decent blokes who usually give me trade discount.
the trouble with a tarp for a Disco is they are so ffing big and thus a bit hard to put on and off, often.
I already have two discos both under tarps, and it is a rigmarole.
Wish I didn't have sun rooffs!