Battery voltage always being the same points to one of two things, the rectifier or the generator.
First thing check your fuses are ok, the feed to the battery from the reg/rec goes through the 20 amp one to get to the battery. A white wire goes from the reg to the fuse and a white/red comes out to the battery. You should check this wire is ok too by unplugging the reg and sticking the meter on 'ping' from the white terminal on the plug to the positive wire on the battery. On my bike there was a bullet connector in the wire near the battery, that small bullet connector was corroded to buggery, all the copper has dissolved in the wire and although it showed continuity it could not carry any current, so bugger all worked without the engine running, except the headlight which was bright.....
The other output from the reg (brown wire) goes and does all the other stuff, powering the cdi, lights etc. While you are by the reg, just check that the earth wire form the frame is actually a wire not a piece of plastic tube masquerading as one. The brown wire is more difficult to check, because it goes off to a bunch of places. So we will leave that for a moment.
Next up is check the generator is working. First resistance, check between the 3 yellow wires, between any 2 of them you should see between 0.3 and 1.0 ohm, if they are outside this then the generator rotor has to come off. Check all three 'pairs'. If one is very different to the others, or any are outside the range, then the generator is buggered
Plug it all back together and with the trusty paper clips (be very careful here do not short the two clips together, you will kill the rotor in very short order) measure the voltage across the pairs of yellow wires. You should see more than 25volts at 4000rpm. Oh have the meter on AC volts, otherwise you will think it is broken and it isn't....
As you are by the plug, turn back to DC volts and measure between the earth point and the white wire and then the brown wire, you should see this voltage get up around the 14volts mark when you rev it on the white one and the brown should be clipped to 12.5 or so. (the white is unregulated, the brown is regulated). If they don't and everything else was ok, the reg is buggered. From what you say in your post I would say the reg is the most likely problem, but doing the simple checks first means you haven't spent any money until you are sure.
There are a bunch of other things that can affect the running, the diode pack can fail and this allows the back emf from the sensor coils into the cdi box which stops it working properly, and then stops it completely.
I don't think you have this issue, because of the battery voltage issue. However if the reg/rec and the wires prove ok, then it will involve further checking.
See how you get on with the generator first.
Oh get a puller, beg borrow or steal one because the rotors weld themselves to the crank, and no amount of hitting, swearing or bodging will get them off. Even with a puller they can be right tricky little bleeders, you may need the trusty blow torch, but only near the puller, not by the magnets.... (The magnets crack very easily, and then the generator does not work properly.) I did take one apart which had about a half of one magnet left.....
You can get them from most bike shops and certainly from M&P or whatever they call themselves these days, and they aren't that expensive. Remember to remove the bolt first. I watched a mate of mine try and pull his off with the bolt in place, still tightened up
After a while I asked why and he said he didn't want to damage the threads....
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