Solar Charge Battery Low Voltage Disconnecting
unless the battery management/charger IC specifically does it, your system load WILL drain your battery to death. I have that issue on my BQ24090 single cell lipo charger i put on my little combat robot. The output of that charger goes to the lipo, but that it also the system load rail! 

You actually need to monitor this issue yourself in most cases: one nice way to do it is monitor the battery voltage using a micro controller (your arduino!) ADC or even an opamp (probably better, then it's not relying on software to do it) which sees that the Lipo is now at 3V (safer to stop here, dont go so low as 2.75V) and then cut off/turn off the system. At this point you assume "the battery is flat" and you deal with it - in your case you would disconnect the load (system, not the lipo) until the battery voltage has been beefed up enough by your solar charge system to turn on the system again. Maybe the "turn back on" voltage is 4V? 

do some investigations into battery monitoring/protection ICs, but basically you need a more complicated one to recover after the solar cells have brought up the battery level enough to continue again. Otherwise you can assume that a human can just replace the battery, and deactivate the "load disconnect" switch, and let the system continue now on a full battery?