I’m no light bulb engineer.
But resistance increases as temp goes up. Which reduces current.
Maybe 6v doesn’t get the temps for a 10w 12v bulb up high enough to limit current and it is in some electronic equation range that kills elements?
All that could be completely wrong. Just what comes to mind. Could try higher and lower watt bulbs to see if they like being on the 6v supply better. I’m basing most of this on someone telling me that household incandescent bulbs don’t last as long when used with a dimmer switch. 🤷🏻♂️