A couple of notes:For power I have a 150 mah as well as a 500 mah, both work with the panel.
It's not good for the LED to have it connected directly to a 3.7V battery (which may be as high as 4.2V when fully charged) as there's no way of limiting the current the LED draws and it will fail prematurely. I'd recommend a 39 or 43 ohm resistor in series with the LED to limit the current to around 20mA when the battery is fully charged.
Your larger LiPo battery has a low voltage cut-off protection circuit built into it that prevents the cell voltage going below 3V, which would cause damage to the cell when it's recharged. The smaller cell does not have a built-in protection circuit, and if it's discharged below 3V you run the risk of damaging it when it's recharged. If you want to use it safely you ideally need an external over-discharge protection module between it and the load.