My B40WD has more than 12.5V at tickover without lights on. On moderate revs it gives more than 13.8 V with lights on.  It's fitted with an A REG ONE rectifier/regulator. Surprises both me and Paul Goff that the old alternator is in so good condition. In my experience the Lucas alternators rarely charge the battery at tickover.

A long time ago, a friend stated that when he had blown a J.A.P. speedway engine to pieces, he heard the engine gremlins shouting Free, free, at last free while running to the woods.

It might be that modern vehicles constantly draws current. Though some modern motorcycles can have a winter sleep and start in spring, other not.
Worst I have is a 2018 Citroen van that has to be run at least once a month to maintain battery charge. Even worse. some modern cars with flat or removed battery must be reprogrammed when fitting a charged battery. Also experienced that some of the modern electronic regulators we fit to our old bikes draw current. I now have the habit to take out the fuse on bikes not frequently used.
As Pete says, your Bonnie battery might need replacement. Or simply sell the bike and rely on the B40.

Edit: Batteries are best stored fully charged and cold. All chemical processes goes slower at low temperatures. A charged battery will not freeze. A poorly charged battery can crack at -30C. Happened to me.

Do the forceps compress the spring and hold it tight to the carburettor top?

Yes, the forceps grips the wire keeping the spring compressed up to the top. Makes it easier to fit wire and needle into the slide.

I've done initial adjustment as Julian said without taking carb off by using a thin welding rod under the throttle slide. Turning the adjusting screw in until the rod can move free. Not certain if it works on a B40 where it's not much space to get a bent rod in.

The 60W E3L don't give much at low revs, so it might be that your battery is draining with lights on. Especially with a 4Ah battery, which I think is less than the original. For a Norton WD16H which had been upgraded with an EL3 from the 40W E3, I fitted LED lamps some 5 years ago. Got no battery drain with lamps on at moderate revs. Back then LED lamp beam pattern was poor, but I'm told that there are LEDs with a decent beam pattern now.

I believe Google isn't always the best source.
According to the owners manual oil capacity is 2.25 litres or i US pints 4 3/4.
Found on https://www.bsaunitsingles.com/owners_repair_manuals.htm