Does battery depletion damage the inverter

I have a small complete 200 watt Renogy system which works well for me in my Michigan cottage. The question is: If I leave the 500 watt inverter on over the winter when I'm gone will the inverter be damaged if the battery gets depleted (which it undoubtedly will in the cloudy winter)). I've noticed I got an alarm this summer when battery depletes - I just turn off the load but would it be a problem if inverter alarms and I'm not there? I would like to run some 5.5 watt LED lights on the solar over the winter if I can.

Thanks