Thanks all - the solar charger would be a problem as there's another garage above me and it would be a bit awkward to get a wire out of the front. I might give the car battery idea a go, as I do have a car to lug it about. I was actually thinking about putting an SAE connector (or similar) in anyway so I could carry something like this around with me:
https://www.amazon.co.uk/DBPOWER-120...dp/B01DLRGY8A/
But then I decided that it was crazy and I should just try and find/fix the problem.
As well as checking the connectors, etc., if I have time I'll check the voltages at the battery and lights this weekend, and I may even see if I can get an ammeter in line (although not when starting).