Non-rechargable Battery health/charge monitoring

I have a miniature datalogger running which is powered by either a small non-rechargable Lithium battery (CR2013) or a set of 3x 1.2V Alkaline AAA cells. A bi-color LED is made to blink green when it is in operation and healthy.

I just need that LED to glow red (& possibly a piezo buzzer start buzzing) when the CR2013 battery is nearing exhaustion and needs to be changed. Can someone help me figure out the circuit that can monitor the health of non-rechargable batteries -- note 2 different chemistries (but in 2 different type of circuits). This monitoring needs to happen in parallel, i.e. while the the battery is powering the data-logger circuit as well.

Is there something within the BQ series (or another series) from TI, which can help me with this ? I am looking at all possible solutions. My application is not mission-critical, but hopefully in 95% cases, this mechanism can detect a battery nearing end-of-charge, and I need this in the most economical fashion, as this is a cost-sensitive application.