The device powers up the first time I plug it in, and will even do the task that requires the high current, no problem with that. The problem is when the arduino goes to sleep, it draws so little power that the battery pack turns off. It turns out it has an auto power off function when there is very low current draw. I believe this exists for when you are charging your phone, that it doesn't just trickle charge itself dead after it has charged the phone.
So my question is, is there an all in one solution out there that you have used that doesn't have the auto off? Is there a better way to do this easily? I was attracted to the all in one units because they are more outdoor friendly. At this point I just need something that works and is fairly cheap. I think Voltaic Systems V44 battery has an "always on" function specifically for applications like this, but they are really expensive.
If you want to find a solution to the "auto off" then IMHO the first step would be to take the charger apart and start looking at the ICs they're using for charge control. There's more than likely some method of changing that shutoff time.
As an alternative, or for future reference, the Seeeduino Stalker is an Arduino with a built-in solar charger and RTC.
Most of these Solar Chargers that produce 5V from Lithium Ion batteries use a Boost Converter to step the voltage up from 3.7 to 5V, and the reason for the shutdown when no current is being drawn is that Boost Converters are quite inefficient at very low loads, so leaving the device running would slowly drain its own battery.
You could try and pull the device apart and see if you can disable the auto shutoff , but you would then need to measure the standby current drain and figure out how long the battery will last.
The Solar Panel would be at best 2 watts given the size of the device.
I have seen some of these power supplies, that are timed. If you will turn the arduino on ever 10 or 15 minutes for a couple milli seconds, the power supply will recognize a load, and keep running.