It's entirely possible that the current bank is shot, although 11.6 volts sounds high for an inverter low cutoff to me (a heavy burst of load with the batteries partially drained could definitely get down to 11.6).
If you want to find out for sure, do a C/20 discharge test. Get the batteries fully charged, then apply a load at the 20 hour rate (60 amps for a 1200ah bank) and time how long it takes to reach 10.5 volts (typical end of discharge voltage at C/20 for lead acid batteries). If it takes much under 20 hours, they've lost some capacity. If you hit 10.5 volts after, say, 10 hours, then they've lost half their capacity and would be considered end of life (usually the threshold to condemn is somewhere around 70 - 80% of original capacity).