With a semi-stable load, and semi-stable solar, I still see fluctuations.
Or with no load, the battery voltage will start to drop when the charging amps are reduced... but still charging. Meaning the voltage can go down even when charging.
Not sure a 1/100th Volt voltage change will tell what you want to know with any accuracy at all.
If you just want a beeper to sound when the voltage drops below a certain threshold, that's pretty easy.
For the meter- Could series 10 AA nicds or a 7AH SLA, charged from a 1.5W HF PV, 7809 voltage regulator, to a $3 HF meter. It works 24/7. Simple. Fairly cheap.
G-