There are alot of things it could be. That means you can guess and spent alot of money before you get it. I suggest that you begin a process of narrowing down the cause, which is what you must do when there are so many possibilities. I would start with finding out if the cause is fuel or spark. Too much fuel? Too little fuel? No spark? Spark, but at the wrong time? F.I. systems can also have a wrong time fuel component, but that usually coincides with wrong time for spark too.
What you describe seems to relate to a cold bike, rather than a hot one. What you describe seems to indicate that once she has warmed up, she is better. Firing back out the intake could be a clue. Sensors can be thermally sensitive. Coils can be thermally sensitive. Wires can be broken inside insulation, and insulation gets softer when warm, allowing their wires to connect or disconnect. The possibilities go on and on.
It could be so many things. Start with: is it fuel or spark? and work your way back from there. Maybe you have more symptoms. Nothing is insignificant. Does she cut out on bumps? Does she hesitate when you accelerate? Does she pop and sputter when taking off from stop? Whatever you see, or hear, or feel, or smell...
Also, 11.7v is considered a fully discharged battery. She may have some things work and some things not at that voltage.
My 2 cents.
Good luck,
Rich P