I believe you have two problems working together...
One, 12v is too low for the battery if you're trying to e-start it!
Either your battery is cooked because of a bad regulator/rectifier OR the stator isn't outputting enough AC (maybe it has one burnt phase ).
It could be even what Bandit127
said...I'm not familiar with ténere stators...
Other, your sparkplug coil could be bad!
It can run fine when cold but when it's warm it'll shut off completely..
or even the sparkplug cap be slightly loose and vibration shakes it of?