So, after many years of pretty faultless service, my 2016 PCX 125 has suddenly stopped starting.

Problem definitely seems electrical. I turn the key, and although the display backlight turns on, sometimes the display doesn't display anything on the LCD, sometimes it does. But when I try to start the engine, I just get a little click noise.
I also have a worry that the fuel meter, when it does show, isn't accurate, as the last time I was out on the PCX, I seemed to drive an unusually long way while the bars stay on 3 bars - I also have a slight worry I've run out of fuel because of this

The headlights still do come on, though, strangely.
I figure either water/damp has got into the display system (not sure that would stop it starting, though), or the battery is on it's way out - I guess at 7 years, it's had a very decent innings, if so.
Based on the "symptoms" what do you guys think it's most likely to be?
To test my dying battery theory, and as it's the cheapest option, I've ordered a 12v charger off Amazon.
Extra question on this. If I connect the 12v charger to the battery, do I need to first disconnect the battery from the bike? Could I frazzle the bike electrics if I don't (I haven't checked yet, but maybe it's physically impossible, anyway?)? Should I maybe also physically remove the battery from the bike before charging, first, just to be on the safe side?
Thanks!