I'd put money on it being the battery. Output voltage is only one consideration, it's when the battery is placed under load that it will prove itself. Try checking the battery voltage with it connected to the panel and the mains disconnected. If it's still reading above 12 volts, it's the panel although I'm fairly sure after 4 years, the battery has had it and the voltage will plummet.
At a guess (not a big Texecom user personally) your bell box has tried to do a battery test and tripped the fault output (due to battery not connected) which you appear to have connected to the global tamper terminals on the control panel? AFAIK, the Aux/Fault loop defaults to a tamper circuit, hence the reason for the full alarm condition. Faults shouldn't cause the alarm to go into full alarm.
If you want the bell to be monitored for faults, wait until you've connected the bell box battery, connect the fault terminals to a spare zone, program the zone as fault or suchlike, and you'll avoid the full alarm condition if it goes into fault in future.
