I think I have an electrical problem :(

No, not whilst charging, whilst trying to start the engine.
But, just to confuse things, earlier in the day whilst trying to start the engine I was getting a voltage dip down to 3v.

My battery has charged up overnight and is giving me 13.25v on the meter, yes I know that this will drop a bit and it doesn't indicate what will happen when it is under load but I don't think the battery is the root of this.

I suspect I have dislodged a bad connection whilst I was in at the engine changing the air filter (which is a reach-around-the-back job). The symptoms I am getting before and after the filter change are so different. Before, the engine was turning over at what seemed like a good speed, no clicking, voltage steady at 12.5; after, even the lights on the panel aren't coming on, let along the engine actually turn over. Yet the battery was still giving me 12.5 the whole time.

I was hoping that since the engine control panel green light is on, yet everything else is dead, that the fault might be narrowed down to a particular part of the wiring.
The control panel is the basic 'A' type, seen here: http://www.betamarine.co.uk/seagoing/sg_controlpanels_options.html
 
I think you're in for some tedious contortionism! You don't say exactly where you're measuring these voltages, though. If at the battery terminals, then 3v is rather bad news. By the way, don't switch battery 2 into the circuit until the problem is sorted or you'll just discharge into battery 1. You could be right that you've distrubed a dodgy connection while worling on adjacent items. You really need to be prgmatic and undo each connection in turm, slean it and remake it. It would be a good idea to measure voltage across each connection as you go to identify improvemnets, but while you're upside down and hugging the engine block you may as well do them all in one hit. Whilst you're doing that, keep charging the battery!

Good luck,
Rob.
 
The big clue has to be where you say:
"What has me really perplexed is that the engine was turning over like a goodun whilst giving me about 12.6v, and on the next try, a couple of hours later (after changing the filters) it would do nothing at all. Well, it would happily power the cabin lights, including the voltage sensitive LEDs, but I've never before had the lights on the engine control panel fail to come on, or the buzzer fail to sound."
And if battery voltage is still good with ignition on, but not cranking, then you need to progressively check from the battery, along the circuit towards the panel lights until you find what is likely to be a bad connection. If cranking is also a problem, then likely to be nearer the battery, eg, isolator, starter solenoid connection, or any intermediate connections. Good luck, where are you based, there may be a boater nearby who can help trace the fault?
 
And an update:
Took the healthy battery home, charged it up, hooked it up to the boat and engine still dead. All other electrics looked very healthy, cabin lights etc, and as before the green light on the engine control panel was lit up but the rest was dead.
The reason? Tracing the wiring back from the panel, I discovered a big chunky wiring connector wrapped in rubber. It had come apart, but I hadn't seen that because of the rubber sleeving. It was in near the air filter so obviously I had dislodged it when changing that out.
Plugged the cable back together, and whoomph, engine up and running.
So, if you have a Beta, and ever find that the green light comes on but nothing else, check the big chunky connection from the rest of the panel- don't blame the battery!
 
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