I think that you're on the right track with a recording type of device. Indeed, since the problem may be intermitant -say, only on Saturday morning- you'ld want to record for a greater period than one night.
BUT- since we have no idea what's going on, why don't we start from scratch?
Verify the volts into the units. Then, verify the volts arriving at the batteries. It isn't impossible that the units themselves are killing the batteries.
Be alert of over-voltage as well as under-voltage. Heat is a real battery killer- and feeding a battery too many volys will kill it quickly.
Indeed, you might place a thermometer in the units. If these are tamper-resistant fixtures, they could very well be too hot. Especially if the night crew covers them up so as to sleep better! I think medical thermometers have a 'stop' that holds the high reading.
Some places have folks with way too much time on their hands. Some bored folks might be cycling the test buttons- either just for fun, or for some light after 'lights out.' Either case, repeated charge cycles probably aren't good for the batteries.
Absent a proper chart recorder, you're left to improvising. I'd wire a small clock, or electromagnet, inside one of the units. These will at least indicate (depending on how wired) if the test button was pressed or the power failed at some point.
I really can't think of an over-voltage indicator, though.
