I misunderstood. I assumed you were doing all this with a charged battery. Are you sure your alternator hook-up isn't draining the battery? Obviously the alternator is charging or the machine would not run with a stone-dead battery. So the question in my mind is - why isn't the current that is suppling the ignition coil making its way to the battery to charge it?
Seems either your alternator output is not connected to the battery - or you've got a bad battery - or your alternator is draining it after shut-down.
How are you getting current to the #1 exciter on the alternator? You cannot wire it directly to the IGN terminal on the key-switch unless you stick in some resistance or a diode. You CAN run it to the ACC terminal which is isolated from the IGN when the key is off. I'm assume that part is working OK if the engine shuts off when you turn the key off.