Sless,
This project is a good use case for PWM. I would agree your LED uses 10watts continuously. There are other loads like the controller, internal resistance of the battery, etc. The 50 watt panel upgrade and the 35 AH battery upgrade will also do nicely. If your controller is shutting off your system at 10.4 volts, you are hurting that battery if it is LA. Other chemistries will vary.
About your calculations for energy produced, you will not get full rating for 10 hours. Full rating is based on 1000watts per square meter which is only seen around high noon. Also, PWM controllers do not capture rated wattage EVER. It is just not possible. So your 20W panel will make at most 15watts using quick math. From there, it goes down earlier and later in the day. Expect 50% for 8 hours. It will be a lot closer to your actual production.
I will be interested to see how the newer setup works. I would also adjust the protection voltage to something closer to 12.2 volts so as to not damage the battery. With a 35 AH battery, you should be able to see 10 hours of use out of that 10 watt light.You will be burning closer to 15 AHs during that time leaving you with more than 50% SOC remaining.
