I've had a battery that had those exact symptoms.
It wasn't long for this earth after that....
Can you try boosting the battery by jumper leads, say from a car.
This would say for sure if its just a low battery, or if you really do have a problem.
Be ready for a quick disconnect of the hot lead from the boost battery...
btw, you say your battery "returns to 12v".
12 point how many volts, precisely ?
A battery at 12.0 volts only is already rather dead...
hopethishelps.