Voltage Drop When Ignition Turned On

Any way you can hook an amp meter ? Sounds to me like you have a short circuit, finding it is going to be interesting

I would start disconnecting stuff and suspect anything that gets really hot , even a wire, so it can lead you to the problem

 

If the battery reads 12v with no load then it's in a low state of charge. A min of 12.6v when fully charged and 14-14.5 with the engine running.
Every circuit, other than the starter motor, runs through a fuse of some sort. The sort of fault which could completely drain a battery in under a minute would blow a fuse long before that! I would expect the ignition circuit to draw no more than a few amps.

 

Had a thing some years ago with my ZZR1200. Started blowing 20A fuse and finished up coming home on a truck from France. Bastard Frogs were too busy whining about the shit World Cup they were having....anyway turned out to be an aftermarket screen l fitted was causing high resistance in a loom in the nosecone. Didn't even cut the wiring or insulation, just trapped it, but turning on was instant death for the fuse. If l've read you correctly the battery does recover so l'd go with a parasitic drain linked to a live circuit. Odd it doesn't take the fuse though.

 

Shorted chaffed wires near the headstock maybe?

Charge your battery up as has been pointed out, 12v is no good, 12.6v is.