Hello Goucho,
I was looking at your stall current of 6.9 amps and thinking you may be able to get around 50% of that.
Lets look at some numbers. I am going to round things off so we have even numbers, but you can tweak them however you see fit.
First of all, yes, you will have to do some testing to see how well things work out.
Your LED current is 350 mA. By the time you add a circuit or resistor, you will most likely be drawing around 500 mA from the battery pack. You want a 20 minute run time, so you are looking at around 170 mAh of capacity from the battery.
Let's look at a charge rate of 1000 mA. Charging has losses, so you will probably need to generate around 200 mAh to get 170 mAh out. 200/1000 = 0.2 hours of charge time, or 12 minutes.
If you wanted to charge the 1200 mAh battery up from empty, you would need to generate around 1440 mAh of capacity. If you are capable of 1000 mA, it would take you around 86 minutes to do a full charge.
Tom