A few point to consider here:-
Your multimeter may not be as accurate as you think, I will assume been a DIY that you haven't splashed out a few hundred quid on it and get it regularly calibrated however.. for the response we will assume the multimeter is accurate as well as its batteries are not low.
A fully charged healthy battery would show around 12.6v +, if you are seeing 12v it would suggest your battery is on its way out or the charging circuit is failing, a fault on the panel or wiring also could drag the voltage down too so take that into consideration.
You definitely should replace the battery as a precaution and you'll be looking around 10-20 quid depending on its size, this may not resolve the issue but from experience some panels do go into tamper fault when the battery is duff so try it, batteries in alarms should be replaced every 4 yrs, it is a good idea to date the battery when you replace it so in future you can recall exactly when it was fitted.